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Parikh vectors
1KRV_1 6PFQ_1 6ICP_1 Letter Amino acid
2 2 9 C Cysteine
9 31 28 L Leucine
7 7 6 M Methionine
9 3 14 Y Tyrosine
8 15 28 A Alanine
0 11 14 Q Glutamine
14 15 25 E Glutamic acid
10 12 14 K Lycine
12 19 32 S Serine
22 20 24 V Valine
16 10 15 N Asparagine
8 6 18 H Histidine
17 6 18 I Isoleucine
8 9 20 F Phenylalanine
12 11 25 P Proline
2 1 8 W Tryptophan
16 12 13 G Glycine
9 7 26 D Aspartic acid
12 5 12 T Threonine
10 16 20 R Arginine 

1KRV_1|Chains A, B, C|GALACTOSIDE O-ACETYLTRANSFERASE|Escherichia coli (562)
>6PFQ_1|Chains A, B, C|Uncharacterized protein YLR132C|Kluyveromyces marxianus (strain DMKU3-1042 / BCC 29191 / NBRC 104275) (1003335)
>6ICP_1|Chains A, B, C|Methylxanthine N1-demethylase NdmA|Pseudomonas putida (303)
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 \(w=f(c)\)
1KRV , Knot 95 203 0.83 38 148 194 
MNMPMTERIRAGKLFTDMCEGLPEKRLRGKTLMYEFNHSHPSEVEKRESLIKEMFATVGENAWVEPPVYFSYGSNIHIGRNFYANFNLTIVDDYTVTIGDNVLIAPNVTLSVTGHPVHHELRKNGEMYSFPITIGNNVWIGSHVVINPGVTIGDNSVIGAGSIVTKDIPPNVVAAGVPCRVIREINDRDKHYYFKDYKVESSV
6PFQ , Knot 96 218 0.79 40 138 204 
MVAPVSKNIGFLFLELRLDSKQQQIMDLVLKGVNAVMDTHHRNSFEPLHRGKFGAMKPLHVSLSETMMFANESELEEKMGRIRQEIRALECKSVPVALSGGWLVYENFDASLQFLAVGLSEPARGRLKPVLSIVEKYKPRSPVSRQPVGLNNLHVSFGVAQNAYLQQDESVSRQRLDSLRNLVATEASDRLPLLRANLQFRCHELKAKVGTSVITLPL
6ICP , Knot 160 369 0.85 40 223 350 
MGSSHHHHHHENLYFQGSMEQAIINDEREYLRHFWHPVCTVTELEKAHPSSLGPLAVKLLNEQLVVAKLGDEYVAMRDRCAHRSAKLSLGTVSGNRLQCPYHGWQYDTHGACQLVPACPNSPIPNKAKVDRFDCEERYGLIWIRLDSSFDCTEIPYFSAANDPRLRIVIQEPYWWDATAERRWENFTDFSHFAFIHPGTLFDPNNAEPPIVPMDRFNGQFRFVYDTPEDMAVPNQAPIGSFSYTCSMPFAINLEVSKYSSSSLHVLFNVSCPVDSHTTKNFLIFAREQSDDSDYLHIAFQDLVLAEDKPVIESQWPKDAPADEVSVVADKVSIQYRKWLRELKEAHKEGSQAFRSALLDPVIESDRSYI

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 \(c\).

\(|P_{f(1KRV_1)}(2) \setminus P_{f(6PFQ_1)}(2)|=90\), \(|P_{f(6PFQ_1)}(2) \setminus P_{f(1KRV_1)}(2)|=80\). 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 1:10111000101101100100111000101001100100001001000001100111011001110111010010010110010101010110000101100111110101010101100010001010011101100111100111011101100011111011000111011111110011001000000001000010001
Pair \(Z_2\) Length of longest common subsequence
1KRV_1,6PFQ_1 170 3
1KRV_1,6ICP_1 179 3
6PFQ_1,6ICP_1 177 4

Newick tree

 
[
	6ICP_1:90.29,
	[
		1KRV_1:85,6PFQ_1:85
	]:5.29
]

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 \((0.85)(0.8)(\frac{421 }{\log_{20} 421}-\frac{203}{\log_{20}203})=64.0\)
Status Protein1 Protein2 d d1/2
Query variables 1KRV_1 6PFQ_1 81 78
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]