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Parikh vectors
7WNT_1 8TLR_1 7RGE_1 Letter Amino acid
33 3 10 P Proline
57 10 16 G Glycine
56 11 24 L Leucine
34 14 14 D Aspartic acid
31 13 19 E Glutamic acid
16 3 8 H Histidine
31 11 15 T Threonine
2 0 6 W Tryptophan
64 15 19 V Valine
42 11 9 R Arginine
13 4 8 N Asparagine
12 11 9 Q Glutamine
11 8 17 K Lycine
16 5 8 F Phenylalanine
31 8 18 S Serine
50 11 20 A Alanine
5 4 1 C Cysteine
6 9 10 Y Tyrosine
35 11 14 I Isoleucine
13 4 1 M Methionine 

7WNT_1|Chain A|Ribonuclease J|Mycobacterium tuberculosis H37Rv (83332)
>8TLR_1|Chain A|GTPase HRas|Homo sapiens (9606)
>7RGE_1|Chain A|3'-phosphoadenylylsulfate reductase|Candida auris (498019)
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 \(w=f(c)\)
7WNT , Knot 216 558 0.81 40 233 514 
MDVDLPPPGPLTSGGLRVTALGGINEIGRNMTVFEHLGRLLIIDCGVLFPGHDEPGVDLILPDMRHVEDRLDDIEALVLTHGHEDHIGAIPFLLKLRPDIPVVGSKFTLALVAEKCREYRITPVFVEVREGQSTRHGVFECEYFAVNHSTPDALAIAVYTGAGTILHTGDIKFDQLPPDGRPTDLPGMSRLGDTGVDLLLCDSTNAEIPGVGPSESEVGPTLHRLIRGADGRVIVACFASNVDRVQQIIDAAVALGRRVSFVGRSMVRNMRVARQLGFLRVADSDLIDIAAAETMAPDQVVLITTGTQGEPMSALSRMSRGEHRSITLTAGDLIVLSSSLIPGNEEAVFGVIDALSKIGARVVTNAQARVHVSGHAYAGELLFLYNGVRPRNVMPVHGTWRMLRANAKLAASTGVPQESILLAENGVSVDLVAGKASISGAVPVGKMFVDGLIAGDVGDITLGERLILSSGFVAVTVVVRRGTGQPLAAPHLHSRGFSEDPKALEPAVRKVEAELESLVAANVTDPIRIAQGVRRTVGKWVGETYRRQPMIVPTVIEV
8TLR , Knot 80 166 0.82 38 131 163 
MTEYKLVVVGACGVGKSALTIQLIQNHFVDEYDPTIEDSYRKQVVIDGETCLLDILDTAGQEEYSAMRDQYMRTGEGFLCVFAINNTKSFEDIHQYREQIKRVKDSDDVPMVLVGNKCDLAARTVESRQAQDLARSYGIPYIETSAKTRQGVEDAFYTLVREIRQH
7RGE , Knot 112 246 0.83 40 165 240 
GSISLSQEHIDHLNKTLVELSPQEVLRWAVVTFPNLYQTTAFGLTGLVILDMISKTKPVDLIFIDTLHHFPQTYDLVRKVAAAYQPTLHIYKPKGVESEEEFAKKHGDSLWESNDDLYDFLVKVEPAQRAYKELGVNAVLTGRRKSQGGARAALPVIEVEESSGIIKINPLWNWDFAQVKAYITENAVPYNELLDLGYKSIGDWHSTVAVADGEDERSGRWKGKAKTECGIHETSRFAQYLASTSS

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 \(c\).

\(|P_{f(7WNT_1)}(2) \setminus P_{f(8TLR_1)}(2)|=137\), \(|P_{f(8TLR_1)}(2) \setminus P_{f(7WNT_1)}(2)|=35\). 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 1:101011111110011101011111001100101100110111100111111000111011110100100010010111100100001111111101010111110010111110000000101111010010000011100001110000101111110011101100101010011101010011110011001101110000010111111000011101001101101011110110010010011011111100101110011001011001111011000110111100111001111001001011011001001000010101101111000111100011111101100111011001010101010101101111001101001111010101101010111001110001111001101011110101011111101110111110110101100111001111101110010101111101000110001011011100101010011110100110110110001101110000001111101101
Pair \(Z_2\) Length of longest common subsequence
7WNT_1,8TLR_1 172 3
7WNT_1,7RGE_1 166 4
8TLR_1,7RGE_1 162 3

Newick tree

 
[
	7WNT_1:85.65,
	[
		7RGE_1:81,8TLR_1:81
	]:4.65
]

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 \((0.85)(0.8)(\frac{724 }{\log_{20} 724}-\frac{166}{\log_{20}166})=157.\)
Status Protein1 Protein2 d d1/2
Query variables 7WNT_1 8TLR_1 194 125
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]