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Parikh vectors
4QNW_1 7XPF_1 8IEG_1 Letter Amino acid
25 12 7 E Glutamic acid
26 30 3 G Glycine
29 41 11 L Leucine
20 16 6 Q Glutamine
19 20 7 I Isoleucine
14 15 6 K Lycine
16 24 3 Y Tyrosine
9 28 1 N Asparagine
11 9 2 M Methionine
17 20 4 F Phenylalanine
23 42 4 P Proline
6 8 0 W Tryptophan
26 36 5 V Valine
45 48 9 A Alanine
26 36 14 R Arginine
20 31 4 D Aspartic acid
5 6 2 C Cysteine
16 8 2 H Histidine
23 47 3 S Serine
16 43 5 T Threonine 

4QNW_1|Chain A|Chanoclavine-I aldehyde reductase|Aspergillus fumigatus (330879)
>7XPF_1|Chains A, B, C|Capsid protein alpha|Lake Sinai virus 2 (1041831)
>8IEG_1|Chains A[auth E], E[auth K]|Histone H3.1|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 \(w=f(c)\)
4QNW , Knot 171 392 0.86 40 228 377 
MREEPSSAQLFKPLKVGRCHLQHRMIMAPTTRFRADGQGVPLPFVQEYYGQRASVPGTLLITEATDITPKAMGYKHVPGIWSEPQREAWREIVSRVHSKKCFIFCQLWATGRAADPDVLADMKDLISSSAVPVEEKGPLPRALTEDEIQQCIADFAQAARNAINAGFDGVEIHGANGYLIDQFTQKSCNHRQDRWGGSIENRARFAVEVTRAVIEAVGADRVGVKLSPYSQYLGMGTMDELVPQFEYLIAQMRRLDVAYLHLANSRWLDEEKPHPDPNHEVFVRVWGQSSPILLAGGYDAASAEKVTEQMAAATYTNVAIAFGRYFISTPDLPFRVMAGIQLQKYDRASFYSTLSREGYLDYPFSAEYMALHNFPVASGKLAAALEHHHHHH
7XPF , Knot 204 520 0.81 40 246 484 
MNPPTTTTTTTRTIRAPKVQLTPNSATRRRRNRRRRRPAAAIAGPLSIQPSSINRRMVSRVTRRSAVVTAAGLAWLRQYLNPMGPDTTSVTGYPDGSAVTTCIADYSNTFNVSFPPREALYCTGSSSSEKPTLVDADNYAKIDKWSNYDITLCVLALPMLRNVVMLRLYPHTPTAFALTEQTPNFPQRFPNWSVYSADGTRFNNGDEPGYLQSYVYLPNVDKHLSAARGYRLLSRGITGIFSAPALETQGFVTACQYLAEGSIQSQSIKSDAVRSVTVNSDGTVKNVESSSQTVSSMPRYVFPLDGDNCAPSSLTETYHQAYQSKATDGFYMPVLSSSRDNPFHPPQPRAIAVYGSFLARGCLDPVSEAHEADGPTHDIYRLNVADDVAPLFNTGVVWFEGISPKFSLKLKTRTVLQYIPTSGSVLANFTRHEPTYDQIALDAADRLRNLMPHAYPAAYNDWGWLGDLLDSAISMLPGVGTVYNIAKPLIKPAWNWLGNKVSDFFGNPVARDGDIFFDAK
8IEG , Knot 52 98 0.81 38 86 95 
KPHRYRPGTVALREIRRYQKSTELLIRKLPFQRLVREIAQDFKTDLRFQSSAVMALQEACEAYLVGLFEDTNLCAIHAKRVTIMPKDIQLARRIRGER

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 \(c\).

\(|P_{f(4QNW_1)}(2) \setminus P_{f(7XPF_1)}(2)|=76\), \(|P_{f(7XPF_1)}(2) \setminus P_{f(4QNW_1)}(2)|=94\). 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 1:10001001011011011000100011111000101010111111100001001011101110010010101110001111100100011001100100000111001110101101011101001100011110001111011000010001101101100110111011010110101100100000000000111010001011101001110111100111010100001111010011101001110100101101011000110000101010001110111000111111100110100100011110000111111001100101110111110100000101000100010100110100111001111010111110000000
Pair \(Z_2\) Length of longest common subsequence
4QNW_1,7XPF_1 170 4
4QNW_1,8IEG_1 192 4
7XPF_1,8IEG_1 204 3

Newick tree

 
[
	8IEG_1:10.30,
	[
		4QNW_1:85,7XPF_1:85
	]:18.30
]

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 \((0.85)(0.8)(\frac{912 }{\log_{20} 912}-\frac{392}{\log_{20}392})=138.\)
Status Protein1 Protein2 d d1/2
Query variables 4QNW_1 7XPF_1 174 152.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]