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Parikh vectors
4XPC_1 6HAV_1 2LWI_1 Letter Amino acid
16 41 11 A Alanine
14 8 11 R Arginine
11 28 11 I Isoleucine
13 17 10 S Serine
4 9 3 C Cysteine
16 24 13 G Glycine
7 8 3 H Histidine
38 24 12 L Leucine
4 14 4 M Methionine
8 11 5 F Phenylalanine
15 16 10 T Threonine
11 25 15 V Valine
7 10 4 N Asparagine
15 19 15 D Aspartic acid
9 29 13 E Glutamic acid
13 26 8 K Lycine
23 21 4 P Proline
6 1 0 W Tryptophan
8 6 9 Y Tyrosine
21 5 11 Q Glutamine 

4XPC_1|Chain A|reverse transcriptase|Moloney murine leukemia virus (isolate Shinnick) (928306)
>6HAV_1|Chain A|5,10-methenyltetrahydromethanopterin hydrogenase|Methanococcus aeolicus (strain ATCC BAA-1280 / DSM 17508 / OCM 812 / Nankai-3) (419665)
>2LWI_1|Chain A|GTPase HRas|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 \(w=f(c)\)
4XPC , Knot 120 259 0.85 40 173 253 
GSHMTWLSDFPQAWAETGGMGLAVRQAPLIIPLKATSTPVSIKQYPMSQEARLGIKPHIQRLLDQGILVPCQSPWNTPLLPVKKPGTNDYRPVQDLREVNKRVEDIHPTVPNPYNLLSGLPPSHQWYTVLDLKDAFFCLRLHPTSQPLFAFEWRDPEMGISGQLTWTRLPQGFKNSPTLFDEALHRDLADFRIQHPDLILLQYVDDLLLAATSELDCQQGTRALLQTLGNLGYRASAKKAQICQKQVKYLGYLLKEGQR
6HAV , Knot 142 342 0.80 40 190 320 
MKVAILGAGCYRSHAACGITNFSRAAEVANKVGIPEITMTHSTITMGAELLHLVDEIDEVVVSDPCFAEEPGLIIIDEFDCKEVMEAHLAGKAEDVMPAIRDAVKAKAKDSPKPPKGCIHFVNPEKVGLKVTSDDREAIEGADIVITWLPKGGSQPAIIEKFVDAIKEGAIVTHACTIPTPKFAKIFKDLGREDLNIVSFHPGCVPEMKGQVYLSEGYASEEAVEKLYKIAKISRGTAFKMPANLISPVCDMGSAVTAPVYAAILSYRDAVTNILGAPADFAQMMADEAITQMLELMRNEGIQNMENKLNPGALTGTADSMCFGPLSELLPASLKVLEEHKK
2LWI , Knot 83 172 0.82 38 136 169 
GPLGSDMTEYKLVVVGAGGVGKSALTIQLIQNHFVDEYDPSIEDSYRKQVVIDGETCLLDILDTAGQEEYSAMRDQYMRTGEGFLCVFAINNTKSFEDIHQYREQIKRVKDSDDVPMVLVGNKCDLAARTVESRQAQDLARSYGIPYIETSAKTRQGVEDAFYTLVREIRQH

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 \(c\).

\(|P_{f(4XPC_1)}(2) \setminus P_{f(6HAV_1)}(2)|=77\), \(|P_{f(6HAV_1)}(2) \setminus P_{f(4XPC_1)}(2)|=94\). 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 1:1001011001101110011111110011111110100011010001100010111010100110011111000110011111001100000110010010001001010110100110111100010011010011101010100011111010010111010101001101100010110011000110101001011110010011111000100001001110011011001010010100001001101100100
Pair \(Z_2\) Length of longest common subsequence
4XPC_1,6HAV_1 171 3
4XPC_1,2LWI_1 155 4
6HAV_1,2LWI_1 174 3

Newick tree

 
[
	6HAV_1:88.97,
	[
		4XPC_1:77.5,2LWI_1:77.5
	]:11.47
]

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 \((0.85)(0.8)(\frac{601 }{\log_{20} 601}-\frac{259}{\log_{20}259})=96.3\)
Status Protein1 Protein2 d d1/2
Query variables 4XPC_1 6HAV_1 123 109
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]