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Parikh vectors
1DZY_1 1ZHW_1 8UYN_1 Letter Amino acid
15 23 10 I Isoleucine
24 45 15 L Leucine
9 23 11 P Proline
6 22 9 D Aspartic acid
4 3 2 C Cysteine
6 13 11 Q Glutamine
11 10 2 H Histidine
14 19 10 T Threonine
10 15 10 N Asparagine
17 44 9 E Glutamic acid
13 22 10 G Glycine
9 42 16 K Lycine
4 24 11 F Phenylalanine
7 13 10 Y Tyrosine
15 18 13 V Valine
10 14 7 R Arginine
5 4 2 M Methionine
10 49 13 S Serine
2 10 2 W Tryptophan
24 25 5 A Alanine 

1DZY_1|Chain A[auth P]|L-FUCULOSE-1-PHOSPHATE ALDOLASE|ESCHERICHIA COLI (562)
>1ZHW_1|Chain A|KES1 protein|Saccharomyces cerevisiae (4932)
>8UYN_1|Chains A, B, C|Neutrophil gelatinase-associated lipocalin|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 \(w=f(c)\)
1DZY , Knot 103 215 0.85 40 156 210 
MERNKLARQIIDTCLEMTRLGLNQGTAGNVSVRYQDGMLITPTGIPYEKLTESHIVFIDGNGKHEEGKLPSSEWRFHMAAYQSRPDANAVVHNHAVHCTAVSILNRSIPAIHYMIAAAGGNSIPCAPYATFGTRELSEHVALALKNRKATLLQHHGLIACEVNLEKALWLAHEVEVLAQLYLTTLAITDPVPVLSDEEIAVVLEKFKTYGLRIAE
1ZHW , Knot 177 438 0.82 40 220 403 
GAMDPSQYASSSSWTSFLKSIASFNGDLSSLSAPPFILSPISLTEFSQYWAEHPELFLEPSFINDDNYKEHCLIDPEVESPELARMLAVTKWFISTLKSQYCSRNESLGSEKKPLNPFLGELFVGKWENKEHPEFGETVLLSEQVSHHPPVTAFSIFNDKNKVKLQGYNQIKASFTKSLMLTVKQFGHTMLDIKDESYLVTPPPLHIEGILVASPFVELEGKSYIQSSTGLLCVIEFSGRGYFSGKKNSFKARIYKDSKDSKDKEKALYTISGQWSGSSKIIKANKKEESRLFYDAARIPAEHLNVKPLEEQHPLESRKAWYDVAGAIKLGDFNLIAKTKTELEETQRELRKEEEAKGISWQRRWFKDFDYSVTPEEGALVPEKDDTFLKLASALNLSTKNAPSGTLVGDKEDRKEDLSSIHWRFQRELWDEEKEIVL
8UYN , Knot 86 178 0.83 40 131 171 
QDSTSDLIPAPPLSKVPLQQNFQDNQFQGKWYVVGLAGNAILREDKDPQKMYATIYELKEDKSYNVTSVLFRKKKCDYWIRTFVPGSQPGEFTLGNIKSYPGLTSYLVRVVSTNYNQHAMVFFKKVSQNREYFKITLYGRTKELTSELKENFIRFSKSLGLPENHIVFPVPIDQCIDG

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 \(c\).

\(|P_{f(1DZY_1)}(2) \setminus P_{f(1ZHW_1)}(2)|=59\), \(|P_{f(1ZHW_1)}(2) \setminus P_{f(1DZY_1)}(2)|=123\). 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 1:10000110011000101001110010110101000011110101110001000011110101000010110001010111000010101110001100011011000111100111111100110110101100010001111100001011000111100101001111100101110101001110011111000011111001000110110
Pair \(Z_2\) Length of longest common subsequence
1DZY_1,1ZHW_1 182 4
1DZY_1,8UYN_1 173 4
1ZHW_1,8UYN_1 173 4

Newick tree

 
[
	1ZHW_1:89.52,
	[
		1DZY_1:86.5,8UYN_1:86.5
	]:3.02
]

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 \((0.85)(0.8)(\frac{653 }{\log_{20} 653}-\frac{215}{\log_{20}215})=123.\)
Status Protein1 Protein2 d d1/2
Query variables 1DZY_1 1ZHW_1 152 114
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]