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Parikh vectors
7URL_1 2HUH_1 5QJQ_1 Letter Amino acid
5 1 2 W Tryptophan
17 10 7 Y Tyrosine
20 14 16 A Alanine
12 5 9 R Arginine
10 8 3 F Phenylalanine
5 3 4 M Methionine
34 6 11 S Serine
6 7 6 N Asparagine
2 2 4 H Histidine
13 11 13 K Lycine
4 0 5 C Cysteine
9 8 13 P Proline
13 15 15 V Valine
45 5 14 G Glycine
11 5 12 I Isoleucine
17 14 17 L Leucine
19 8 16 T Threonine
9 10 12 D Aspartic acid
12 3 7 Q Glutamine
14 12 23 E Glutamic acid 

7URL_1|Chains A, B|anti-HIV scFv|Mus musculus (10090)
>2HUH_1|Chain A|Putative DNA mismatch repair protein|Bacteroides thetaiotaomicron (818)
>5QJQ_1|Chains A, B, C, D|ADP-sugar pyrophosphatase|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 \(w=f(c)\)
7URL , Knot 113 277 0.76 40 156 239 
MAEVKLVESGGGLVKPGGSLKLSCAASGFTFSRFGMHWVRQTPEKRLEWVAYISSGSSTIYYADTVKGRFTISRDNAKNTLYLQMSSLRSEDTAIYYCARSGGIERYDGTYYVMDYWGQGTTLTVSSGGGGSGGGGSGGGGSDIVLTQSPASLTVSLGQRATISCRASESVDYYGKSFMNWYQQKPGQPPKLLIYAASNQGSGIPARFSGSGSGTDFTLNIHPVEEEDAATYYCQQSKEVPWTFGAGTKLEIKRAAAKGEFGGGGSGGGGSENLYFQ
2HUH , Knot 73 147 0.82 38 115 144 
RQPEVRGGDTLNVFLAYVPEDAKAMMTTPFEAYLVNDSNYYLYYTYLSAEGKAWNNRSHGLVEPNTKLLLEEFTKDVLNEMERVAVQLIAFKDGKPAAIKPAVSVELRIDTVKFYKLHTFSASDFFEEPALIYDIVKDDVPAKQVYV
5QJQ , Knot 97 209 0.82 40 143 202 
SMESQEPTESSQNGKQYIISEELISEGKWVKLEKTTYMDPTGKTRTWESVKRTTRKEQTADGVAVIPVLQRTLHYECIVLVKQFRPPMGGYCIEFPAGLIDDGETPEAAALRELEEETGYKGDIAECSPAVCMDPGLSNCTIHIVTVTINGDDAENARPKPKPGDGEFVEVISLPKNDLLQRLDALVAEEHLTVDARVYSYALALKHAN

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 \(c\).

\(|P_{f(7URL_1)}(2) \setminus P_{f(2HUH_1)}(2)|=95\), \(|P_{f(2HUH_1)}(2) \setminus P_{f(7URL_1)}(2)|=54\). 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 1:1101011001111101110101001101101001110110001000101110100100010010010101010000100010101001000001100010011100001000110011010010100111101111011110011100011010101100101000100010001001101000011011011101100010111101010101001010101100001100000000011101111001010011101011111011110001010
Pair \(Z_2\) Length of longest common subsequence
7URL_1,2HUH_1 149 4
7URL_1,5QJQ_1 157 3
2HUH_1,5QJQ_1 156 3

Newick tree

 
[
	5QJQ_1:79.46,
	[
		7URL_1:74.5,2HUH_1:74.5
	]:4.96
]

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 \((0.85)(0.8)(\frac{424 }{\log_{20} 424}-\frac{147}{\log_{20}147})=82.7\)
Status Protein1 Protein2 d d1/2
Query variables 7URL_1 2HUH_1 95 75
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]