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Parikh vectors
2KOZ_1 4EMU_1 6ACE_1 Letter Amino acid
1 7 21 V Valine
4 14 18 R Arginine
0 14 20 E Glutamic acid
0 4 9 H Histidine
1 9 10 I Isoleucine
2 6 15 T Threonine
0 1 4 W Tryptophan
0 8 19 P Proline
1 4 12 K Lycine
4 14 15 S Serine
1 11 5 Y Tyrosine
1 27 18 L Leucine
1 2 3 M Methionine
1 15 30 A Alanine
2 9 8 N Asparagine
3 11 7 D Aspartic acid
6 4 9 C Cysteine
1 14 7 Q Glutamine
4 9 24 G Glycine
0 5 13 F Phenylalanine 

2KOZ_1|Chain A|nasonin-1|synthetic construct (32630)
>4EMU_1|Chains A, B|Transmembrane protein 173|Homo sapiens (9606)
>6ACE_1|Chain A|NAD-dependent protein deacylase sirtuin-5, mitochondrial|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 \(w=f(c)\)
2KOZ , Knot 21 33 0.74 30 29 30 
ACNDRDCSLDCIMKGYNTGSCVRGSCQCRRTSG
4EMU , Knot 88 188 0.81 40 133 185 
SVAHGLAWSYYIGYLRLILPELQARIRTYNQHYNNLLRGAVSQRLYILLPLDCGVPDNLSMADPNIRFLDKLPQQTGDHAGIKDRVYSNSIYELLENGQRAGTCVLEYATPLQTLFAMSQYSQAGFSREDRLEQAKLFCRTLEDILADAPESQNNCRLIAYQEPADDSSFSLSQEVLRHLRQEEKEEV
6ACE , Knot 121 267 0.84 40 181 262 
PSSSMADFRKFFAKAKHIVIISGAGVSAESGVPTFRGAGGYWRKWQAQDLATPLAFAHNPSRVWEFYHYRREVMGSKEPNAGHRAIAECETRLGKQGRRVVVITQNIDELHRKAGTKNLLEIHGSLFKTRCTSCGVVAENYKSPICPALSGKGAPEPGTQDASIPVEKLPRCEEAGCGGLLRPHVVWFGENLDPAILEEVDRELAHCDLCLVVGTSSVVYPAAMFAPQVAARGVPVAEFNTETTPATNRFRFHFQGPCGTTLPEALA

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 \(c\).

\(|P_{f(2KOZ_1)}(2) \setminus P_{f(4EMU_1)}(2)|=18\), \(|P_{f(4EMU_1)}(2) \setminus P_{f(2KOZ_1)}(2)|=122\). 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 1:100000001001101000100101000000001
Pair \(Z_2\) Length of longest common subsequence
2KOZ_1,4EMU_1 140 3
2KOZ_1,6ACE_1 186 2
4EMU_1,6ACE_1 180 4

Newick tree

 
[
	6ACE_1:97.63,
	[
		2KOZ_1:70,4EMU_1:70
	]:27.63
]

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 \((0.85)(0.8)(\frac{221 }{\log_{20} 221}-\frac{33}{\log_{20}33})=64.1\)
Status Protein1 Protein2 d d1/2
Query variables 2KOZ_1 4EMU_1 84 48
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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