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Parikh vectors
4POM_1 9FNZ_1 5ODW_1 Letter Amino acid
0 4 43 R Arginine
6 0 59 D Aspartic acid
7 7 39 E Glutamic acid
7 4 72 G Glycine
6 1 61 L Leucine
9 3 59 A Alanine
9 2 31 F Phenylalanine
3 6 35 P Proline
5 3 57 T Threonine
11 4 43 V Valine
3 2 49 N Asparagine
5 0 32 Q Glutamine
1 2 14 H Histidine
16 3 37 K Lycine
5 8 0 C Cysteine
4 4 39 I Isoleucine
3 2 15 M Methionine
7 2 65 S Serine
1 2 18 W Tryptophan
1 1 47 Y Tyrosine 

4POM_1|Chains A, B, C, D|Thioredoxin|Homo sapiens (9606)
>9FNZ_1|Chains A[auth A1], FA[auth B1]|RNA-binding protein|Pyrococcus furiosus (2261)
>5ODW_1|Chains A, B|Ferripyoverdine receptor|Pseudomonas aeruginosa (strain ATCC 15692 / DSM 22644 / CIP 104116 / JCM 14847 / LMG 12228 / 1C / PRS 101 / PAO1) (208964)
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 \(w=f(c)\)
4POM , Knot 56 109 0.80 38 86 106 
GAGTMVKQIESKTAFQKALKAAGDKLVVVDFSATWCGPCKMIKPFFHSLSEKYSNVIFLEVDVDDCQDVASECEVKCMPTFQFFKKGQKVGEFSGANKKKLEATINKLV
9FNZ , Knot 33 60 0.75 36 51 55 
MVEMKFEIPVCTSCGREITPREHATHFICPNCGEAVIWRCESCRVLAKPYKCPNCGWEGP
5ODW , Knot 310 815 0.85 38 292 746 
MPAPHGLSPLSKAFLMRRAFQRRILPHSLAMALSLPLAGYVQAQEVEFDIPPQALGSALQEFGRQADIQVLYRPEEVRNKRSSAIKGKLEPNQAITELLRGTGASVDFQGNAITISVAEAADSSVDLGATMITSNQLGTITEDSGSYTPGTIATATRLVLTPRETPQSITVVTRQNMDDFGLNNIDDVMRHTPGITVSAYDTDRNNYYARGFSINNFQYDGIPSTARNVGYSAGNTLSDMAIYDRVEVLKGATGLLTGAGSLGATINLIRKKPTHEFKGHVELGAGSWDNYRSELDVSGPLTESGNVRGRAVAAYQDKHSFMDHYERKTSVYYGILEFDLNPDTMLTVGADYQDNDPKGSGWSGSFPLFDSQGNRNDVSRSFNNGAKWSSWEQYTRTVFANLEHNFANGWVGKVQLDHKINGYHAPLGAIMGDWPAPDNSAKIVAQKYTGETKSNSLDIYLTGPFQFLGREHELVVGTSASFSHWEGKSYWNLRNYDNTTDDFINWDGDIGKPDWGTPSQYIDDKTRQLGSYMTARFNVTDDLNLFLGGRVVDYRVTGLNPTIRESGRFIPYVGAVYDLNDTYSVYASYTDIFMPQDSWYRDSSNKLLEPDEGQNYEIGIKGEYLDGRLNTSLAYFEIHEENRAEEDALYNSKPTNPAITYAYKGIKAKTKGYEAEISGELAPGWQVQAGYTHKIIRDDSGKKVSTWEPQDQLSLYTSYKFKGALDKLTVGGGARWQGKSWQMVYNNPRSRWEKFSQEDYWLVDLMARYQITDKLSASVNVNNVFDKTYYTNIGFYTSASYGDPRNLMFSTRWDF

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 \(c\).

\(|P_{f(4POM_1)}(2) \setminus P_{f(9FNZ_1)}(2)|=75\), \(|P_{f(9FNZ_1)}(2) \setminus P_{f(4POM_1)}(2)|=40\). 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 1:1110110010000110011011100111101010101100110111001000000111101010000011000010011010110010011010110000101010011
Pair \(Z_2\) Length of longest common subsequence
4POM_1,9FNZ_1 115 2
4POM_1,5ODW_1 242 4
9FNZ_1,5ODW_1 277 4

Newick tree

 
[
	5ODW_1:14.44,
	[
		4POM_1:57.5,9FNZ_1:57.5
	]:88.94
]

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 \((0.85)(0.8)(\frac{169 }{\log_{20} 169}-\frac{60}{\log_{20}60})=37.2\)
Status Protein1 Protein2 d d1/2
Query variables 4POM_1 9FNZ_1 48 36.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]