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Parikh vectors
6PXZ_1 1HZA_1 8RHK_1 Letter Amino acid
15 4 18 A Alanine
0 0 0 C Cysteine
5 5 6 Q Glutamine
3 1 3 H Histidine
3 1 4 M Methionine
5 6 9 F Phenylalanine
5 4 6 N Asparagine
1 1 26 L Leucine
11 2 9 R Arginine
8 1 14 D Aspartic acid
4 9 17 E Glutamic acid
1 3 15 I Isoleucine
4 1 12 P Proline
6 2 23 S Serine
1 3 19 T Threonine
5 1 2 W Tryptophan
3 7 16 V Valine
11 10 23 G Glycine
9 5 18 K Lycine
4 1 10 Y Tyrosine 

6PXZ_1|Chains A, B, C|Serum amyloid A-3 protein|Mus musculus (10090)
>1HZA_1|Chains A, B|COLD SHOCK PROTEIN CSPB|Bacillus caldolyticus (1394)
>8RHK_1|Chains A, O|Proteasome subunit alpha type-2|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
6PXZ , Knot 55 104 0.81 38 86 102 
QRWVQFMKEAGQGSRDMWRAYSDMKKANWKNSDKYFHARGNYDAARRGPGGAWAAKVISDAREAVQKFTGHGAEDSRADQFANEWGRSGKDPNHFRPAGLPKRY
1HZA , Knot 38 67 0.79 38 60 65 
MQRGKVKWFNNEKGYGFIEVEGGSDVFVHFTAIQGEGFKTLEEGQEVSFEIVQGNRGPQAANVTKEA
8RHK , Knot 111 250 0.81 38 159 239 
MTDRYSFSLTTFSPSGKLGQIDYALTAVKQGVTSLGIKATNGVVIATEKKSSSPLAMSETLSKVSLLTPDIGAVYSGMGPDYRVLVDKSRKVAHTSYKRIYGEYPPTKLLVSEVAKIMQEATQSGGVRPFGVSLLIAGHDEFNGFSLYQVDPSGSYFPWKATAIGKGSVAAKTFLEKRWNDELELEDAIHIALLTLKESVEGEFNGDTIELAIIGDENPDLLGYTGIPTDKGPRFRKLTSQEINDRLEAL

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 \(c\).

\(|P_{f(6PXZ_1)}(2) \setminus P_{f(1HZA_1)}(2)|=67\), \(|P_{f(1HZA_1)}(2) \setminus P_{f(6PXZ_1)}(2)|=41\). 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 1:00110110011010001101000100101000000101010001100111111110110010011001010110000100110011001001001011111000
Pair \(Z_2\) Length of longest common subsequence
6PXZ_1,1HZA_1 108 3
6PXZ_1,8RHK_1 179 3
1HZA_1,8RHK_1 151 3

Newick tree

 
[
	8RHK_1:90.37,
	[
		6PXZ_1:54,1HZA_1:54
	]:36.37
]

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 \((0.85)(0.8)(\frac{171 }{\log_{20} 171}-\frac{67}{\log_{20}67})=35.2\)
Status Protein1 Protein2 d d1/2
Query variables 6PXZ_1 1HZA_1 44 36
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]