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Parikh vectors
9MOJ_1 3EUI_1 4NXT_1 Letter Amino acid
22 0 54 L Leucine
8 0 13 Y Tyrosine
3 0 32 A Alanine
8 0 21 D Aspartic acid
1 0 7 C Cysteine
3 0 14 Q Glutamine
4 0 23 P Proline
8 0 23 S Serine
5 4 15 T Threonine
9 0 22 V Valine
3 0 9 N Asparagine
5 8 17 G Glycine
1 0 5 H Histidine
0 0 5 W Tryptophan
5 0 23 R Arginine
20 0 21 E Glutamic acid
19 0 15 I Isoleucine
17 0 14 K Lycine
2 0 5 M Methionine
8 0 9 F Phenylalanine 

9MOJ_1|Chains A, B|SsoGINS51|Saccharolobus solfataricus P2 (273057)
>3EUI_1|Chains A, B, C, D|5'-D(*DGP*DGP*DGP*DGP*DTP*DTP*DTP*DTP*DGP*DGP*DGP*DG)-3'|null
>4NXT_1|Chains A, B, C, D|Mitochondrial dynamic protein MID51|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 \(w=f(c)\)
9MOJ , Knot 72 151 0.79 38 103 144 
MLDELVKKELSEEEITEIKLEEIIKYITLIKKSKTFVSSEIRKEELKFLSELAESLFELRLSKVLEGKVGKGFDEFIFDIFKILKQFYVDLLTGRYIIYNDKIYCIVQKPLIYNDHRVNEGDVLVLPMREALPLIIASYLTPYKIDIEEQL
3EUI , Knot 4 12 0.27 4 4 6 
GGGGTTTTGGGG
4NXT , Knot 150 347 0.84 40 195 334 
GSARKGQVDLKKSRLRMSLQEKLLTYYRNRAAIPAGEQARAKQAAVDICAELRSFLRAKLPDMPLRDMYLSGSLYDDLQVVTADHIQLIVPLVLEQNLWSCIPGEDTIMNVPGFFLVRRENPEYFPRGSSYWDRCVVGGYLSPKTVADTFEKVVAGSINWPAIGSLLDYVIRPAPPPEALTLEVQYERDKHLFIDFLPSVTLGDTVLVAKPHRLAQYDNLWRLSLRPAETARLRALDQADSGCRSLCLKILKAICKSTPALGHLTASQLTNVILHLAQEEADWSPDMLADRFLQALRGLISYLEAGVLPSALNPKVNLFAELTPEEIDELGYTLYCSLSEPEVLLQT

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 \(c\).

\(|P_{f(9MOJ_1)}(2) \setminus P_{f(3EUI_1)}(2)|=102\), \(|P_{f(3EUI_1)}(2) \setminus P_{f(9MOJ_1)}(2)|=3\). 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 1:1100110001000010010100110010110000011000100001011001100110101001101011011001110110110010101101001100001001100111000001001011111100111111100101001010001
Pair \(Z_2\) Length of longest common subsequence
9MOJ_1,3EUI_1 105 2
9MOJ_1,4NXT_1 166 4
3EUI_1,4NXT_1 197 2

Newick tree

 
[
	4NXT_1:10.70,
	[
		9MOJ_1:52.5,3EUI_1:52.5
	]:48.20
]

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 \((0.85)(0.8)(\frac{163 }{\log_{20} 163}-\frac{12}{\log_{20}12})=55.3\)
Status Protein1 Protein2 d d1/2
Query variables 9MOJ_1 3EUI_1 70 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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]