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Parikh vectors
3LYF_1 4NUI_1 5ODW_1 Letter Amino acid
17 4 43 R Arginine
11 7 59 D Aspartic acid
1 12 0 C Cysteine
9 5 31 F Phenylalanine
11 5 35 P Proline
8 7 47 Y Tyrosine
33 22 59 A Alanine
15 15 39 E Glutamic acid
14 22 72 G Glycine
4 3 14 H Histidine
23 13 61 L Leucine
12 13 15 M Methionine
15 11 43 V Valine
10 10 32 Q Glutamine
8 4 39 I Isoleucine
3 2 18 W Tryptophan
11 11 49 N Asparagine
13 17 65 S Serine
11 6 57 T Threonine
16 21 37 K Lycine 

3LYF_1|Chains A, B, C, D|Nucleocapsid protein|Rift Valley fever virus (11588)
>4NUI_1|Chain A|Ancylostoma secreted protein 2|Necator americanus (51031)
>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 \(w=f(c)\)
3LYF , Knot 109 245 0.81 40 163 233 
MDNYQELAIQFAAQAVDRNEIEQWVREFAYQGFDARRVIELLKQYGGADWEKDAKKMIVLALTRGNKPRRMMMKMSKEGKATVEALINKYKLKEGNPSRDELTLSRVAAALAGRTCQALVVLSEWLPVTGTTMDGLSPAYPRHMMHPSFAGMVDPSLPGDYLRAILDAHSLYLLQFSRVINPNLRGRTKEEVAATFTQPMNAAVNSNFISHEKRREFLKAFGLVDSNGKPSAAVMAAAQAYKTAA
4NUI , Knot 95 210 0.80 40 154 200 
MMSSITCLVLLSIAAYSKAGCPDNGMSEEARQKFLEMHNSLRSSVALGQAKDGAGGNAPKAAKMKTMAYDCEVEKTAMNNAKQCVFKHSQPNQRKGLGENIFMSSDSGMDKAKAAEQASKAWFGELAEKGVGQNLKLTGGLFSRGVGHYTQMVWQETVKLGCYVEACSNMCYVVCQYGPAGNMMGKDIYEKGEPCSKCENCDKEKGLCSA
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 \(c\).

\(|P_{f(3LYF_1)}(2) \setminus P_{f(4NUI_1)}(2)|=87\), \(|P_{f(4NUI_1)}(2) \setminus P_{f(3LYF_1)}(2)|=78\). 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 1:10000011101110110000100110011001101001101100011101000100111111001001001110100010101011100001001010000101001111111000011111001111010010110110100110101111101011100101110100101101001101010100000111010011011100011000000011011111000101011111110100011
Pair \(Z_2\) Length of longest common subsequence
3LYF_1,4NUI_1 165 3
3LYF_1,5ODW_1 185 4
4NUI_1,5ODW_1 212 4

Newick tree

 
[
	5ODW_1:10.52,
	[
		3LYF_1:82.5,4NUI_1:82.5
	]:22.02
]

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 \((0.85)(0.8)(\frac{455 }{\log_{20} 455}-\frac{210}{\log_{20}210})=71.4\)
Status Protein1 Protein2 d d1/2
Query variables 3LYF_1 4NUI_1 92 84.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]