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Parikh vectors
4CYW_1 4DNC_1 8XQO_1 Letter Amino acid
33 14 18 G Glycine
16 33 32 K Lycine
7 3 10 M Methionine
8 21 15 Y Tyrosine
17 18 19 V Valine
16 8 31 A Alanine
11 14 31 D Aspartic acid
14 13 14 Q Glutamine
8 11 20 F Phenylalanine
23 7 16 N Asparagine
10 9 6 H Histidine
24 30 28 L Leucine
15 17 29 E Glutamic acid
5 7 3 W Tryptophan
14 16 4 P Proline
29 22 19 S Serine
29 11 21 T Threonine
15 9 18 R Arginine
9 8 10 C Cysteine
21 18 26 I Isoleucine 

4CYW_1|Chains A, C, E|HEMAGGLUTININ|INFLUENZA A VIRUS (A/MALLARD/SWEDEN/51/2002 (H10N2)) (757360)
>4DNC_1|Chains A, B|Histone acetyltransferase KAT8|Homo sapiens (9606)
>8XQO_1|Chain A|Guanine nucleotide-binding protein G(i) subunit alpha-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 \(w=f(c)\)
4CYW , Knot 141 324 0.83 40 204 315 
LDKICLGHHAVANGTIVKTLTNEQEEVTNATETVESTSLDRLCMKGRSHKDLGNCHPIGMLIGTPACDLHLTGTWDTLIERENAIAYCYPGATVNEEALRQKIMESGGISKISTGFTYGSSINSAGTTKACMRNGGNSFYAELKWLVSKSKGQNFPQTTNTYRNTDTAEHLIMWGIHHPSSTQEKNDLYGTQSLSISVGSSTYQSNFVPVVGARPQVNGQSGRIDFHWTLVQPGDNITFSHNGGLIAPSRVSKLIGRGLGIQSDAPIDNNCESKCFWRGGSINTRLPFQNLSPRTVGQCPKYVNKKSLMLATGMRNVPEIVQGR
4DNC , Knot 131 289 0.85 40 185 276 
HEAITKVKYVDKIHIGNYEIDAWYFSPFPEDYGKQPKLWLCEYCLKYMKYEKSYRFHLGQCQWRQPPGKEIYRKSNISVYEVDGKDHKIYCQNLCLLAKLFLDHKTLYFDVEPFVFYILTEVDRQGAHIVGYFSKEKESPDGNNVACILTLPPYQRRGYGKFLIAFSYELSKLESTVGSPEKPLSDLGKLSYRSYWSWVLLEILRDFRGTLSIKDLSQMTSITQNDIISTLQSLNMVKYWKGQHVICVTPKLVEEHLKSAQYKKPPITVDSVCLKWAPPKHKQVKLSKK
8XQO , Knot 159 370 0.84 40 211 350 
MDYKDDDDKENLYFQSMGCTLSAEDKAAVERSKMIDRNLREDGEKAAREVKLLLLGAGESGKSTIVKQMKIIHEAGYSEEECKQYKAVVYSNTIQSIIAIIRAMGRLKIDFGDSARADDARQLFVLAGAAEEGFMTAELAGVIKRLWKDSGVQACFNRSREYQLNDSAAYYLNDLDRIAQPNYIPTQQDVLRTRVKTTGIVETHFTFKDLHFKMFDVGGQRSERKKWIHCFEGVTAIIFCVALSDYDLVLAEDEEMNRMHESMKLFDSICNNKWFTDTSIILFLNKKDLFEEKIKKSPLTICYPEYAGSNTYEEAAAYIQCQFEDLNKRKDTKEIYTHFTCATDTKNVQFVFDAVTDVIIKNNLKDCGLF

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 \(c\).

\(|P_{f(4CYW_1)}(2) \setminus P_{f(4DNC_1)}(2)|=97\), \(|P_{f(4DNC_1)}(2) \setminus P_{f(4CYW_1)}(2)|=78\). 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 1:100101100111010110010000001001000100001001010100000110001111111011001010101001100001110001110100011000110011100100110010010011000101001100101010111000010011000000000001001111110010000000010100010101100000001111111010101001010101011011001010001111110010011101111000111000000001101101000111001010011001001000011110110011011010
Pair \(Z_2\) Length of longest common subsequence
4CYW_1,4DNC_1 175 3
4CYW_1,8XQO_1 187 4
4DNC_1,8XQO_1 182 4

Newick tree

 
[
	8XQO_1:93.79,
	[
		4CYW_1:87.5,4DNC_1:87.5
	]:6.29
]

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 \((0.85)(0.8)(\frac{613 }{\log_{20} 613}-\frac{289}{\log_{20}289})=90.6\)
Status Protein1 Protein2 d d1/2
Query variables 4CYW_1 4DNC_1 114 108
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]