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Parikh vectors
6MCI_1 6ANV_1 5HMV_1 Letter Amino acid
0 10 11 R Arginine
0 26 7 D Aspartic acid
0 10 3 Q Glutamine
0 21 7 T Threonine
20 37 12 G Glycine
0 27 6 I Isoleucine
0 36 7 L Leucine
0 41 6 K Lycine
0 9 2 M Methionine
0 23 2 P Proline
0 30 14 N Asparagine
0 4 1 H Histidine
0 18 3 F Phenylalanine
0 20 10 S Serine
0 9 6 W Tryptophan
0 25 6 V Valine
8 52 12 A Alanine
15 0 8 C Cysteine
0 35 2 E Glutamic acid
0 19 3 Y Tyrosine 

6MCI_1|Chain A|7SK RNA|Homo sapiens (9606)
>6ANV_1|Chains A, B|anti-CRISPR protein AcrF1 fused with C-terminal MBP tag|Pseudomonas phage JBD30 (1223260)
>5HMV_1|Chain A|Lysozyme C|Gallus gallus (9031)
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 \(w=f(c)\)
6MCI , Knot 20 57 0.47 10 17 37 
GGGAUCUGUCACCCCAUUGAUCGCCGAGAGGCUGAUCUGGXUGGXUAGGCGGGUCCC
6ANV , Knot 185 452 0.83 38 220 416 
GSMKFIKYLSTAHLNYMNIAVYENGSKIKARVENVVNGKSVGARDFDSTEQLESWFYGLPGSGLGRIENAMNEISRRENPLEKIEEGKLVIWINGDKGYNGLAEVGKKFEKDTGIKVTVEHPDKLEEKFPQVAATGDGPDIIFWAHDRFGGYAQSGLLAEITPDKAFQDKLYPFTWDAVRYNGKLIAYPIAVEALSLIYNKDLLPNPPKTWEEIPALDKELKAKGKSALMFNLQEPYFTWPLIAADGGYAFKYENGKYDIKDVGVDNAGAKAGLTFLVDLIKNKHMNADTDYSIAEAAFNKGETAMTINGPWAWSNIDTSKVNYGVTVLPTFKGQPSKPFVGVLSAGINAASPNKELAKEFLENYLLTDEGLEAVNKDKPLGAVALKSYEEELAKDPRIAATMENAQKGEIMPNIPQMSAFWYAVRTAVINAASGRQTVDEALKDAQTNAAA
5HMV , Knot 66 128 0.83 40 103 126 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCR

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 \(c\).

\(|P_{f(6MCI_1)}(2) \setminus P_{f(6ANV_1)}(2)|=14\), \(|P_{f(6ANV_1)}(2) \setminus P_{f(6MCI_1)}(2)|=217\). 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 1:111100010010000100110010011111100110001100110011101110000
Pair \(Z_2\) Length of longest common subsequence
6MCI_1,6ANV_1 231 3
6MCI_1,5HMV_1 116 2
6ANV_1,5HMV_1 203 3

Newick tree

 
[
	6ANV_1:12.99,
	[
		6MCI_1:58,5HMV_1:58
	]:62.99
]

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 \((0.85)(0.8)(\frac{509 }{\log_{20} 509}-\frac{57}{\log_{20}57})=137.\)
Status Protein1 Protein2 d d1/2
Query variables 6MCI_1 6ANV_1 181 99.5
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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