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Parikh vectors
7WUZ_1 2PDX_1 7NDT_1 Letter Amino acid
13 13 15 Q Glutamine
16 23 26 E Glutamic acid
13 11 8 F Phenylalanine
6 16 16 S Serine
9 15 22 T Threonine
5 6 9 W Tryptophan
10 19 5 I Isoleucine
26 20 15 P Proline
16 25 15 V Valine
23 19 19 A Alanine
4 15 5 N Asparagine
5 6 5 C Cysteine
12 16 20 G Glycine
26 33 21 L Leucine
3 6 4 M Methionine
10 11 12 Y Tyrosine
17 12 20 R Arginine
12 16 17 D Aspartic acid
8 9 14 H Histidine
7 25 9 K Lycine 

7WUZ_1|Chain A|Cyanocobalamin reductase / alkylcobalamin dealkylase|Homo sapiens (9606)
>2PDX_1|Chain A|Aldose reductase|Homo sapiens (9606)
>7NDT_1|Chains A[auth AAA], F[auth FFF]|HLA class I histocompatibility antigen, alpha chain E|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 \(w=f(c)\)
7WUZ , Knot 108 241 0.82 40 170 232 
EFMEPKVAELKQKIEDTLCPFGFEVYPFQVAWYNELLPPAFHLPLPGPTLAFLVLSTPAMFDRALKPFLQSCHLRMLTDPVDQCVAYHLGRVRESLPELQIEIIADYEVHPNRRPKILAQTAAHVAGAAYYYQRQDVEADPWGNQRISGVCIHPRFGGWFAIRGVVLLPGIEVPDLPPRKPHDCVPTRADRIALLEGFNFHWRDWTYRDAVTPQERYSEEQKAYFSTPPAQRLALLGLAQP
2PDX , Knot 141 316 0.85 40 209 305 
MASRILLNNGAKMPILGLGTWKSPPGQVTEAVKVAIDVGYRHIDCAHVYQNENEVGVAIQEKLREQVVKREELFIVSKLWCTYHEKGLVKGACQKTLSDLKLDYLDLYLIHWPTGFKPGKEFFPLDESGNVVPSDTNILDTWAAMEELVDEGLVKAIGISNFNHLQVEMILNKPGLKYKPAVNQIECHPYLTQEKLIQYCQSKGIVVTAYSPLGSPDRPWAKPEDPSLLEDPRIKAIAAKHNKTTAQVLIRFPMQRNLVVIPKSVTPERIAENFKVFDFELSSQDMTTLLSYNRNWRVCALLRDTSHKDYPFHEEF
7NDT , Knot 121 277 0.82 40 181 264 
MGSHSLKYFHTSVSRPGRGEPRFISVGYVDDTQFVRFDNDAASPRMVPRAPWMEQEGSEYWDRETRSARDTAQIFRVNLRTLRGYYNQSEAGSHTLQWMHGCELGPDGRFLRGYEQCAYDGKDYLTLNEDLRSWTAVDTAAQISEQKSNDASEAEHQRAYLEDTCVEWLHKYLEKGKETLLHLEPPKTHVTHHPISDHEATLRCWALGFYPAEITLTWQQDGEGHTQDTELVETRPAGDGTFQKWAAVVVPSGEEQRYTCHVQHEGLPEPVTLRWKP

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 \(c\).

\(|P_{f(7WUZ_1)}(2) \setminus P_{f(2PDX_1)}(2)|=60\), \(|P_{f(2PDX_1)}(2) \setminus P_{f(7WUZ_1)}(2)|=99\). 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 1:0110101101000100010111101011011100011111101111110111111001111001101110000101100110001100110100011010101110001010001011100110111110000000101011100010110101011111110111111110110111001000110010011110110101001000011010000000001010011100111111101
Pair \(Z_2\) Length of longest common subsequence
7WUZ_1,2PDX_1 159 3
7WUZ_1,7NDT_1 171 3
2PDX_1,7NDT_1 184 4

Newick tree

 
[
	7NDT_1:91.70,
	[
		7WUZ_1:79.5,2PDX_1:79.5
	]:12.20
]

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 \((0.85)(0.8)(\frac{557 }{\log_{20} 557}-\frac{241}{\log_{20}241})=89.9\)
Status Protein1 Protein2 d d1/2
Query variables 7WUZ_1 2PDX_1 114 99.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]