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Parikh vectors
5SQY_1 4XOK_1 3TNQ_1 Letter Amino acid
6 7 15 T Threonine
9 5 19 D Aspartic acid
6 7 21 I Isoleucine
5 3 13 P Proline
6 6 26 E Glutamic acid
2 1 6 M Methionine
11 3 17 S Serine
22 4 20 V Valine
15 2 23 A Alanine
16 2 16 N Asparagine
3 6 15 Q Glutamine
19 9 32 L Leucine
7 1 14 Y Tyrosine
2 4 15 R Arginine
3 0 2 C Cysteine
6 1 8 H Histidine
0 0 6 W Tryptophan
13 6 22 G Glycine
13 7 35 K Lycine
5 2 25 F Phenylalanine 

5SQY_1|Chains A, B|Non-structural protein 3|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4XOK_1|Chains A, B, C|Ubiquitin|Homo sapiens (9606)
>3TNQ_1|Chain A[auth B]|Protein kinase, cAMP-dependent, catalytic, alpha|Rattus norvegicus (10116)
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 \(w=f(c)\)
5SQY , Knot 80 169 0.81 38 119 162 
SMVNSFSGYLKLTDNVYIKNADIVEEAKKVKPTVVVNAANVYLKHGGGVAGALNKATNNAMQVESDDYIATNGPLKVGGSCVLSGHNLAKHCLHVVGPNVNKGEDIQLLKSAYENFNQHEVLLAPLLSAGIFGADPIHSLRVCVDTVRTNVYLAVFDKNLYDKLVSSFL
4XOK , Knot 40 76 0.76 36 61 74 
MQIFVKTLTGKTITLEVEPSDTIENVKAKIQDKEGIPPDQQRLIFAGKQLEDGRTLSDYNIQKESTLHLVLRLRGG
3TNQ , Knot 152 350 0.84 40 209 337 
GNAAAAKKGSEQESVKEFLAKAKEDFLKKWETPSQNTAQLDQFDRIKTLGTGSFGRVMLVKHKESGNHYAMKILDKQKVVKLKQIEHTLNEKRILQAVNFPFLVKLEFSFKDNSNLYMVMEYVAGGEMFSHLRRIGRFSEPHARFYAAQIVLTFEYLHSLDLIYRDLKPENLLIDQQGYIQVTDFGFAKRVKGRTWTLCGTPEYLAPEIILSKGYNKAVDWWALGVLIYEMAAGYPPFFADQPIQIYEKIVSGKVRFPSHFSSDLKDLLRNLLQVDLTKRFGNLKNGVNDIKNHKWFATTDWIAIYQRKVEAPFIPKFKGPGDTSNFDDYEEEEIRVSINEKCGKEFTEF

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 \(c\).

\(|P_{f(5SQY_1)}(2) \setminus P_{f(4XOK_1)}(2)|=92\), \(|P_{f(4XOK_1)}(2) \setminus P_{f(5SQY_1)}(2)|=34\). 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 1:0110010101010001010010110010010101110110101001111111100100011010000011001110111001101001100010111101001001011001000100001111111011111101100101010010001011110001000110011
Pair \(Z_2\) Length of longest common subsequence
5SQY_1,4XOK_1 126 2
5SQY_1,3TNQ_1 162 3
4XOK_1,3TNQ_1 180 3

Newick tree

 
[
	3TNQ_1:91.92,
	[
		5SQY_1:63,4XOK_1:63
	]:28.92
]

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 \((0.85)(0.8)(\frac{245 }{\log_{20} 245}-\frac{76}{\log_{20}76})=54.9\)
Status Protein1 Protein2 d d1/2
Query variables 5SQY_1 4XOK_1 67 48
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]