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Parikh vectors
1ZMA_1 5WIN_1 7NQJ_1 Letter Amino acid
5 19 4 N Asparagine
3 7 2 C Cysteine
6 8 3 Q Glutamine
7 33 10 L Leucine
7 22 12 K Lycine
2 4 6 M Methionine
0 4 1 W Tryptophan
8 21 5 E Glutamic acid
2 9 6 H Histidine
9 11 1 T Threonine
8 14 6 V Valine
10 15 11 A Alanine
6 12 6 R Arginine
7 14 4 G Glycine
8 19 3 I Isoleucine
4 14 6 P Proline
3 6 7 Y Tyrosine
7 16 11 D Aspartic acid
8 14 5 F Phenylalanine
8 15 6 S Serine 

1ZMA_1|Chain A|bacterocin transport accessory protein|Streptococcus pneumoniae (1313)
>5WIN_1|Chain A|Tyrosine-protein kinase JAK2|Homo sapiens (9606)
>7NQJ_1|Chain A|Bromodomain-containing protein 2|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 \(w=f(c)\)
1ZMA , Knot 62 118 0.83 38 101 113 
SNAMEQFLDNIKDLEVTTVVRAQEALDKKETATFFIGRKTCPYCRKFAGTLSGVVAETKAHIYFINSEEPSQLNDLQAFRSRYGIPTVPGFVHITDGQINVRCDSSMSAQEIKDFAGL
5WIN , Knot 124 277 0.84 40 185 271 
VFHKIRNEDLIFNESLGQGTFTKIFKGVRREVGDYGQLHETEVLLKVLDKAHRNYSESFFEAASMMSKLSHKHLVLNYGVCVCGDENILVQEFVKFGSLDTYLKKNKNCINILWKLEVAKQLAAAMHFLEENTLIHGNVCAKNILLIREEDRKTGNPPFIKLSDPGISITVLPKDILQERIPWVPPECIENPKNLNLATDKWSFGTTLWEICSGGDKPLSALDSQRKLQFYEDRHQLPAPKAAELANLINNCMDYEPDHRPSFRAIIRDLNSLFTPD
7NQJ , Knot 59 115 0.81 40 94 113 
GSMGKLSEQLKHCNGILKELLSKKHAAYAWPFYKPVDASALGLHDYHDIIKHPMDLSTVKRKMENRDYRDAQEFAADVRLMFSNCYKYNPPDHDVVAMARKLQDVFEFRYAKMPD

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 \(c\).

\(|P_{f(1ZMA_1)}(2) \setminus P_{f(5WIN_1)}(2)|=38\), \(|P_{f(5WIN_1)}(2) \setminus P_{f(1ZMA_1)}(2)|=122\). 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 1:0011001100100101001101001100000101111000010000111010111100010101100001001001011000011101111101001010100000101001001111
Pair \(Z_2\) Length of longest common subsequence
1ZMA_1,5WIN_1 160 3
1ZMA_1,7NQJ_1 141 3
5WIN_1,7NQJ_1 181 4

Newick tree

 
[
	5WIN_1:89.83,
	[
		1ZMA_1:70.5,7NQJ_1:70.5
	]:19.33
]

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 \((0.85)(0.8)(\frac{395 }{\log_{20} 395}-\frac{118}{\log_{20}118})=84.1\)
Status Protein1 Protein2 d d1/2
Query variables 1ZMA_1 5WIN_1 110 76
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]