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Parikh vectors
2HSO_1 4WZN_1 6FNR_1 Letter Amino acid
7 13 26 D Aspartic acid
8 2 2 C Cysteine
8 24 39 L Leucine
3 4 6 Y Tyrosine
11 15 26 R Arginine
14 8 7 N Asparagine
3 5 6 Q Glutamine
2 22 24 E Glutamic acid
12 12 23 G Glycine
1 4 6 H Histidine
2 8 12 P Proline
7 8 20 T Threonine
12 12 46 A Alanine
3 10 13 F Phenylalanine
10 18 16 S Serine
6 10 11 I Isoleucine
2 9 5 M Methionine
6 5 5 W Tryptophan
6 9 23 V Valine
6 19 7 K Lycine 

2HSO_1|Chain A|Lysozyme C|Gallus gallus (9031)
>4WZN_1|Chains A, B|Genome polyprotein|Human hepatitis A virus genotype IB (isolate HM175) (12098)
>6FNR_1|Chains A, B|Histidine N-alpha-methyltransferase|Mycobacterium smegmatis (1772)
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 \(w=f(c)\)
2HSO , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
4WZN , Knot 101 217 0.83 40 151 212 
SMMSRIAAGDLESSVDDPRSEEDKRFESHIECRKPYKELRLEVGKQRLKYAQEELSNEVLPPPRKMKGLFSQAKISLFYTEEHEIMKFSWRGVTADTRALRRFGFSLAAGRSVWTLEMDAGVLTGRLIRLNDEKWTEMKDDKIVSLIEKFTSNKYWSKVNFPHGMLDLEEIAANSKDFPNMSETDLCFLLHWLNPKKINLADRMLGLSGVQEIKEQG
6FNR , Knot 133 323 0.79 40 168 300 
GHMALSLANYLAADSAAEALRRDVRAGLTATQKSLPPKWFYDAVGSDLFDQITRLPEYYPTRTEAQILRTRSAEIISAAGADTLVELGSGTSEKTRMLLDAMRDAELLRRFIPFDVDAGVLRSAGAAIGAEYPGIEIDAVCGDFEEHLGKIPHVGRRLVVFLGSTIGNLTPAPRAEFLSTLADTLQPGDSLLLGTDLVKDTGRLVRAYDDAAGVTAAFNRNVLAVVNRELSADFDLDAFEHVAKWNSDEERIEMWLRARTAQHVRVAALDLEVDFAAGEEMLTEVSCKFRPENVVAELAEAGLRQTHWWTDPAGDFGLSLAVR

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 \(c\).

\(|P_{f(2HSO_1)}(2) \setminus P_{f(4WZN_1)}(2)|=58\), \(|P_{f(4WZN_1)}(2) \setminus P_{f(2HSO_1)}(2)|=105\). 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 1:011100011111000110000100110110110100010001000000100001110100011000100110001001100111000101010010011001011011111000001001011101001
Pair \(Z_2\) Length of longest common subsequence
2HSO_1,4WZN_1 163 4
2HSO_1,6FNR_1 170 3
4WZN_1,6FNR_1 161 5

Newick tree

 
[
	2HSO_1:84.17,
	[
		4WZN_1:80.5,6FNR_1:80.5
	]:3.67
]

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 \((0.85)(0.8)(\frac{346 }{\log_{20} 346}-\frac{129}{\log_{20}129})=66.4\)
Status Protein1 Protein2 d d1/2
Query variables 2HSO_1 4WZN_1 87 68.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]