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Parikh vectors
7ZWJ_1 9IRV_1 4WRK_1 Letter Amino acid
0 3 0 C Cysteine
0 35 15 E Glutamic acid
5 32 17 G Glycine
0 23 3 H Histidine
0 3 1 W Tryptophan
0 17 9 N Asparagine
1 24 8 D Aspartic acid
0 53 12 L Leucine
0 7 2 M Methionine
0 9 4 F Phenylalanine
1 23 15 V Valine
0 13 6 Q Glutamine
1 15 15 I Isoleucine
4 22 10 K Lycine
1 8 6 P Proline
2 24 13 S Serine
0 9 11 T Threonine
0 12 6 Y Tyrosine
0 47 10 A Alanine
2 15 6 R Arginine 

7ZWJ_1|Chain A|Triculamin|Streptomyces triculaminicus (2816232)
>9IRV_1|Chains A, B|DARPin|synthetic construct (32630)
>4WRK_1|Chains A, B, C, D, E, F|DUTPase|Staphylococcus phage phi11 (12360)
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 \(w=f(c)\)
7ZWJ , Knot 11 17 0.61 16 14 15 
SKKSKPGDGIRGKGVRG
9IRV , Knot 150 394 0.75 40 193 333 
MGHHHHHHGPGSRMKQLEDKVEELLSKNYHLENEVARLKKLVGGSGGSGGSELGKELLEAARAGQDDEVAVLMARGAEVNAADDVGVTPLHLAAQRGHLAIVSVLLAFGASVNAADLWGQTPLHLAATAGHLEIVEVLLRSGASVNARDNIGHTPLHLAAWAGHLEIVEVLLAYGADVFAQDKFGKTPFDLAIDNGNEDIAEVLQRLLECRRDAEAAINYQINLELYASYVYLSMSYYFDRDDVALKNFAKYFLHQSHEEREHAEKLMKLQNQRGGEISLQSISSPDSDDWESGLNAMESALHLEKAVNASLLRLHKLATDCNDPHLCDFIETHYLNEQVKAIKELGDHVTNLRKMGAPESGLAEYLFDKHTLGSGSGAEIEQAKKEIAYLIKK
4WRK , Knot 81 169 0.82 38 129 162 
MTNTLQVRLLSENARMPERNHKTDAGYDIFSAETVVLEPQEKAVIKTDVAVSIPEGYVGLLTSRSGVSSKTHLVIETGKIDAGYHGNLGINIKNNAIASNGYITPGVFDIKGEIDLSDAIRQYGTYQINEGDKLAQLVIVPIWTPELKQVEEFESVSERGEKGFGSSGV

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 \(c\).

\(|P_{f(7ZWJ_1)}(2) \setminus P_{f(9IRV_1)}(2)|=7\), \(|P_{f(9IRV_1)}(2) \setminus P_{f(7ZWJ_1)}(2)|=186\). 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 1:00000110110101101
Pair \(Z_2\) Length of longest common subsequence
7ZWJ_1,9IRV_1 193 2
7ZWJ_1,4WRK_1 123 2
9IRV_1,4WRK_1 144 4

Newick tree

 
[
	9IRV_1:91.67,
	[
		7ZWJ_1:61.5,4WRK_1:61.5
	]:30.17
]

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 \((0.85)(0.8)(\frac{411 }{\log_{20} 411}-\frac{17}{\log_{20}17})=126.\)
Status Protein1 Protein2 d d1/2
Query variables 7ZWJ_1 9IRV_1 143 74.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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