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Parikh vectors
5KSI_1 5GRG_1 4RPO_1 Letter Amino acid
4 4 8 N Asparagine
18 16 23 L Leucine
11 8 4 K Lycine
3 14 4 Y Tyrosine
9 21 14 T Threonine
3 25 19 R Arginine
8 21 7 D Aspartic acid
1 4 2 C Cysteine
1 19 8 Q Glutamine
4 23 21 E Glutamic acid
10 9 8 H Histidine
21 22 17 A Alanine
7 7 13 F Phenylalanine
13 13 15 V Valine
7 21 13 G Glycine
0 8 16 I Isoleucine
2 5 2 M Methionine
7 14 15 P Proline
11 12 11 S Serine
1 9 3 W Tryptophan 

5KSI_1|Chains A, C|Hemoglobin subunit alpha|Homo sapiens (9606)
>5GRG_1|Chain A|HLA class I histocompatibility antigen, A-11 alpha chain|Homo sapiens (9606)
>4RPO_1|Chains A, B, C, D|PCP degradation transcriptional activation protein|Sphingobium chlorophenolicum (46429)
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 \(w=f(c)\)
5KSI , Knot 69 141 0.80 38 99 137 
VLSPADKTNVKAAWGKVGAHAGEYGAEALERMFLSFPTTKTYFPHFDLSHGSAQVKGHGKKVADALTNAVAHVDDMPNALSALSDLHAHKLRVDPVNFKLLSHCLLVTLAAHLPAEFTPAVHASLDKFLASVSTVLTSKYR
5GRG , Knot 122 275 0.83 40 183 265 
GSHSMRYFYTSVSRPGRGEPRFIAVGYVDDTQFVRFDSDAASQRMEPRAPWIEQEGPEYWDQETRNVKAQSQTDRVDLGTLRGYYNQSEDGSHTIQIMYGCDVGPDGRFLRGYRQDAYDGKDYIALNEDLRSWTAADMAAQITKRKWEAAHAAEQQRAYLEGRCVEWLRRYLENGKETLQRTDPPKTHMTHHPISDHEATLRCWALGFYPAEITLTWQRDGEDQTQDTELVETRPAGDGTFQKWAAVVVPSGEEQRYTCHVQHEGLPKPLTLRWE
4RPO , Knot 104 223 0.84 40 159 217 
RSELRFDPGTSNRNFRIAASDFGQALMLPRLYATLEETAPQVRVTGVNLRHGPLVEELESGSIDIAFGGFPTLSAGIKTQTLFREEYVCVMRQSHPALTHGLDLEAFRQCRHIIVTAHEFNHVHEQVEARLLELLPPESIRFTTENFLVSAVIAEETDVILTIPSRLARWFANRGGLTIFPVPIELPSIEVKQYWHERYDKDPGNIWLRRVIAKIGFQNPPAE

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 \(c\).

\(|P_{f(5KSI_1)}(2) \setminus P_{f(5GRG_1)}(2)|=52\), \(|P_{f(5GRG_1)}(2) \setminus P_{f(5KSI_1)}(2)|=136\). 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 1:110110000101111011101100110110011101100000110101001010101010011011001110100110110110010100101011010110001110111011101011101010011101001100000
Pair \(Z_2\) Length of longest common subsequence
5KSI_1,5GRG_1 188 4
5KSI_1,4RPO_1 162 3
5GRG_1,4RPO_1 178 4

Newick tree

 
[
	5GRG_1:94.78,
	[
		5KSI_1:81,4RPO_1:81
	]:13.78
]

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 \((0.85)(0.8)(\frac{416 }{\log_{20} 416}-\frac{141}{\log_{20}141})=82.4\)
Status Protein1 Protein2 d d1/2
Query variables 5KSI_1 5GRG_1 108 79.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]