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Parikh vectors
1WHN_1 6RFZ_1 3VRL_1 Letter Amino acid
12 15 6 R Arginine
9 13 8 E Glutamic acid
11 33 18 G Glycine
9 22 15 L Leucine
2 7 5 F Phenylalanine
2 24 5 W Tryptophan
3 4 12 Y Tyrosine
7 39 13 A Alanine
6 31 22 V Valine
4 0 5 C Cysteine
0 11 2 H Histidine
3 1 5 M Methionine
11 16 13 P Proline
6 30 23 T Threonine
5 19 9 D Aspartic acid
14 37 35 S Serine
8 11 5 Q Glutamine
6 23 4 I Isoleucine
9 5 12 K Lycine
1 30 7 N Asparagine 

1WHN_1|Chain A|hypothetical protein RIKEN cDNA 2310016K04|Mus musculus (10090)
>6RFZ_1|Chain A|lectin PLL2 from Photorhabdus laumondii|Photorhabdus laumondii subsp. laumondii TTO1 (243265)
>3VRL_1|Chains A[auth E], D[auth H]|A10F9 Fab heavy chain|Mus musculus (10090)
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 \(w=f(c)\)
1WHN , Knot 65 128 0.82 38 103 121 
GSSGSSGSGIIKMAIRFDRRAYPPQITPKMCLLEWCRREKLPQPVYETVQRTIDRMFCSVVTVAEQKYQSTLWDKSKKLAEQTAAIVCLRSQGLPEGRLGEESPSLNKRKREAPDQDPGGPRSGPSSG
6RFZ , Knot 112 371 0.59 38 141 233 
MQEEPTNIPRPDNAELLVASEVAIENAAIALSEIVSVVNTSDGRIEVFGVGTDNAVWHNRQTAPHSGSSWTGWISLNGKVTSKPVVYINTDGRLEVFARGTDNALWHIWQTATNAGWSNWQSLGGTITSNPAVYVNTDGRIDVFARGTDNALWHISQTAAHSGPWSSWQSLNGVITSNPAVHINSDGRLEVFARGTDNALWHIWQTAPDSNQWSGWDSLGGVITSDPVVIGTADGRLEVFARGSNNALYHIWQTVPHGGPWSNWASLNGVITSAPAVVKNSDGRLEVFARGTNNALYHIWQTVSHSGPWSNWATLNGTITSAPTAVEDADGRLEVFARGTDNALWNIWQATPSWSAWVSLKGSLIDASAIK
3VRL , Knot 101 224 0.81 40 145 208 
DVKLVESGGGLVKPGGSLKLSCAASGFTFSSYTMSWVRQTPEKRLEWVAIISSGGSYTYYSDSVKGRFTISRDNAKNTLYLQMSSLKSEDTAMYYCTRDEGNGNYVEAMDYWGQGTSVTVSSAKTTPPSVYPLAPGSAAQTNSMVTLGCLVKGYFPEPVTVTWNSGSLSSGVHTFPAVLQSDLYTLSSSVTVPSSTWPSETVTCNVAHPASSTKVDKKIVPRDC

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 \(c\).

\(|P_{f(1WHN_1)}(2) \setminus P_{f(6RFZ_1)}(2)|=61\), \(|P_{f(6RFZ_1)}(2) \setminus P_{f(1WHN_1)}(2)|=99\). 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 1:10010010111011101000101101010101101000001101100010001001100110110000000110000011000111101000111010110001010000001100011110011001
Pair \(Z_2\) Length of longest common subsequence
1WHN_1,6RFZ_1 160 4
1WHN_1,3VRL_1 150 3
6RFZ_1,3VRL_1 154 3

Newick tree

 
[
	6RFZ_1:79.65,
	[
		1WHN_1:75,3VRL_1:75
	]:4.65
]

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 \((0.85)(0.8)(\frac{499 }{\log_{20} 499}-\frac{128}{\log_{20}128})=109.\)
Status Protein1 Protein2 d d1/2
Query variables 1WHN_1 6RFZ_1 98 73
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]