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Parikh vectors
4IGU_1 2QQT_1 1NYJ_1 Letter Amino acid
6 37 2 A Alanine
12 38 1 R Arginine
8 14 1 H Histidine
9 27 5 I Isoleucine
2 14 1 W Tryptophan
5 28 2 V Valine
1 15 0 C Cysteine
5 27 0 Q Glutamine
8 31 0 F Phenylalanine
7 36 0 N Asparagine
6 32 2 D Aspartic acid
4 40 1 G Glycine
10 33 0 K Lycine
4 11 0 M Methionine
6 41 1 P Proline
15 32 3 S Serine
7 27 0 T Threonine
4 15 0 Y Tyrosine
15 32 0 E Glutamic acid
14 65 6 L Leucine 

4IGU_1|Chains A, B|CG5036|Drosophila melanogaster (7227)
>2QQT_1|Chain A|Lactoperoxidase|Bos taurus (9913)
>1NYJ_1|Chains A, B, C, D|Matrix protein M2|null
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 \(w=f(c)\)
4IGU , Knot 73 148 0.82 40 115 143 
MRGSHHHHHHGSQPTLEEIRSWGKSFDKLMKSTAGRKVFQNFLRSEFSEENILFWLACEDLKKENSPELVEEKARLIYEDYISILSPREVSLDSRVREIVNRNMIEPTTHTFDEAQIQIYTLMHRDSYPRFLNSQKFKTLSRPAAKLN
2QQT , Knot 234 595 0.83 40 282 550 
SWEVGCGAPVPLVKCDENSPYRTITGDCNNRRSPALGAANRALARWLPAEYEDGLALPFGWTQRKTRNGFRVPLAREVSNKIVGYLDEEGVLDQNRSLLFMQWGQIVDHDLDFAPETELGSNEHSKTQCEEYCIQGDNCFPIMFPKNDPKLKTQGKCMPFFRAGFVCPTPPYQSLAREQINAVTSFLDASLVYGSEPSLASRLRNLSSPLGLMAVNQEAWDHGLAYLPFNNKKPSPCEFINTTARVPCFLAGDFRASEQILLATAHTLLLREHNRLARELKKLNPHWNGEKLYQEARKILGAFIQIITFRDYLPIVLGSEMQKWIPPYQGYNNSVDPRISNVFTFAFRFGHMEVPSTVSRLDENYQPWGPEAELPLHTLFFNTWRIIKDGGIDPLVRGLLAKKSKLMNQDKMVTSELRNKLFQPTHKIHGFDLAAINLQRCRDHGMPGYNSWRGFCGLSQPKTLKGLQTVLKNKILAKKLMDLYKTPDNIDIWIGGNAEPMVERGRVGPLLACLLGRQFQQIRDGDRFWWENPGVFTEKQRDSLQKVSFSRLICDNTHITKVPLHAFQANNYPHDFVDCSTVDKLDLSPWASREN
1NYJ , Knot 16 25 0.68 22 22 23 
SSDPLVVAASIIGILHLILWILDRL

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 \(c\).

\(|P_{f(4IGU_1)}(2) \setminus P_{f(2QQT_1)}(2)|=23\), \(|P_{f(2QQT_1)}(2) \setminus P_{f(4IGU_1)}(2)|=190\). 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 1:1010000000100101001001100100110001100110011000100001111110001000001011000101100001011010010100010011000110100001001010100110000010110000100100111010
Pair \(Z_2\) Length of longest common subsequence
4IGU_1,2QQT_1 213 4
4IGU_1,1NYJ_1 123 2
2QQT_1,1NYJ_1 272 4

Newick tree

 
[
	2QQT_1:13.49,
	[
		4IGU_1:61.5,1NYJ_1:61.5
	]:74.99
]

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 \((0.85)(0.8)(\frac{743 }{\log_{20} 743}-\frac{148}{\log_{20}148})=168.\)
Status Protein1 Protein2 d d1/2
Query variables 4IGU_1 2QQT_1 213 130.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]