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Parikh vectors
6JED_1 2GIL_1 6BMY_1 Letter Amino acid
12 6 25 N Asparagine
2 8 24 Q Glutamine
17 9 44 E Glutamic acid
24 13 39 L Leucine
13 1 19 P Proline
6 0 19 H Histidine
25 11 40 K Lycine
11 14 34 T Threonine
8 6 20 Y Tyrosine
9 12 30 D Aspartic acid
6 2 7 W Tryptophan
8 9 18 F Phenylalanine
20 11 30 S Serine
12 10 19 A Alanine
3 12 31 R Arginine
1 0 8 C Cysteine
20 8 39 G Glycine
10 11 25 I Isoleucine
0 4 12 M Methionine
21 15 43 V Valine 

6JED_1|Chain A|Metallo-beta-lactamase type 2|Serratia marcescens (615)
>2GIL_1|Chains A, B, C, D|Ras-related protein Rab-6A|Homo sapiens (9606)
>6BMY_1|Chains A, B|Tyrosine-protein phosphatase non-receptor type 11|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 \(w=f(c)\)
6JED , Knot 104 228 0.82 38 147 216 
AESLPDLKIEKLDEGVYVHTSFEEVNGWGVVPKHGLVVLVNAEAYLIDTPFTAKDTEKLVTWFVERGYKIKGSISSHFHSDSTGGIEWLNSRSIPTYASELTNELLKKDGKVQATNSFSGVNYWLVKNKIEVFYPGPGHTPDNVVVWLPERKILFGGCFIKPYGLGNLGDANIEAWPKSAKLLKSKYGKAKLVVPSHSEVGDASLLKLTLEQAVKGLNESKKPSKPSN
2GIL , Knot 79 162 0.82 36 128 159 
KFKLVFLGEQSVGKTSLITRFMYDSFDNTYQATIGIDFLSKTMYLEDRTVRLQLWDTAGQERFRSLIPSYIRDSTVAVVVYDITNVNSFQQTTKWIDDVRTERGSDVIIMLVGNKTDLADKRQVSIEEGERKAKELNVMFIETSAKAGYNVKQLFRRVAAAL
6BMY , Knot 211 526 0.83 40 259 482 
SMTSRRWFHPNITGVEAENLLLTRGVDGSFLARPSKSNPGDFTLSVRRNGAVTHIKIQNTGDYYDLYGGEKFATLAELVQYYMEHHGQLKEKNGDVIELKYPLNCADPTSERWFHGHLSGKEAEKLLTEKGKHGSFLVRESQSHPGDFVLSVRTGDDKGESNDGKSKVTHVMIRCQELKYDVGGGERFDSLTDLVEHYKKNPMVETLGTVLQLKQPLNTTRINAAEIESRVRELSKLAETTDKVKQGFWEEFETLQQQECKLLYSRKEGQRQENKNKNRYKNILPFDHTRVVLHDGDPNEPVSDYINANIIMPEFETKCNNSKPKKSYIATQGCLQNTVNDFWRMVFQENSRVIVMTTKEVERGKSKCVKYWPDEYALKEYGVMRVRNVKESAAHDYTLRELKLSKVGQGNTERTVWQYHFRTWPDHGVPSDPGGVLDFLEEVHHKQESIMDAGPVVVHCSAGIGRTGTFIVIDILIDIIREKGVDCDIDVPKTIQMVRSQRSGMVQTEAQYRFIYMAVQHYIETL

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 \(c\).

\(|P_{f(6JED_1)}(2) \setminus P_{f(2GIL_1)}(2)|=91\), \(|P_{f(2GIL_1)}(2) \setminus P_{f(6JED_1)}(2)|=72\). 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 1:100110101001001101000100101111110011111101010110011010000011011100100101010001000001110110000110010010001100010101000101100111000101101111001001111110001111101101011101101010111001011000010101111000011010110101001101100000100100
Pair \(Z_2\) Length of longest common subsequence
6JED_1,2GIL_1 163 3
6JED_1,6BMY_1 182 4
2GIL_1,6BMY_1 199 3

Newick tree

 
[
	6BMY_1:99.53,
	[
		6JED_1:81.5,2GIL_1:81.5
	]:18.03
]

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 \((0.85)(0.8)(\frac{390 }{\log_{20} 390}-\frac{162}{\log_{20}162})=68.2\)
Status Protein1 Protein2 d d1/2
Query variables 6JED_1 2GIL_1 87 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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]