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Parikh vectors
8GUT_1 7MNL_1 2BFO_1 Letter Amino acid
21 26 21 T Threonine
13 28 11 Y Tyrosine
31 50 43 A Alanine
6 25 8 H Histidine
4 27 16 P Proline
26 35 10 I Isoleucine
9 12 6 M Methionine
26 42 13 D Aspartic acid
10 17 7 C Cysteine
13 34 4 Q Glutamine
27 104 26 L Leucine
18 61 20 S Serine
18 37 16 R Arginine
15 31 12 N Asparagine
30 56 8 K Lycine
19 21 6 F Phenylalanine
3 13 2 W Tryptophan
19 44 25 V Valine
28 60 12 E Glutamic acid
18 30 22 G Glycine 

8GUT_1|Chain A|Guanine nucleotide-binding protein G(i) subunit alpha-1|Homo sapiens (9606)
>7MNL_1|Chains A, B|E3 SUMO-protein ligase RanBP2|Homo sapiens (9606)
>2BFO_1|Chains A, B, C, D|PTERIDINE REDUCTASE 1|LEISHMANIA MAJOR (5664)
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 \(w=f(c)\)
8GUT , Knot 153 354 0.84 40 206 335 
MGCTLSAEDKAAVERSKMIDRNLREDGEKAAREVKLLLLGAGESGKSTIVKQMKIIHEAGYSEEECKQYKAVVYSNTIQSIIAIIRAMGRLKIDFGDSARADDARQLFVLAGAAEEGFMTAELAGVIKRLWKDSGVQACFNRSREYQLNDSAAYYLNDLDRIAQPNYIPTQQDVLRTRVKTTGIVETHFTFKDLHFKMFDVGGQRSERKKWIHCFEGVTAIIFCVALSDYDLVLAEDEEMNRMHESMKLFDSICNNKWFTDTSIILFLNKKDLFEEKIKKSPLTICYPEYAGSNTYEEAAAYIQCQFEDLNKRKDTKEIYTHFTCATDTKNVQFVFDAVTDVIIKNNLKDCGLF
7MNL , Knot 287 753 0.84 40 286 686 
SMRRSKADVERYIASVQGSTPSPRQKSMKGFYFAKLYYEAKEYDLAKKYICTYINVQERDPKAHRFLGLLYELEENTDKAVECYRRSVELNPTQKDLVLKIAELLCKNDVTDGRAKYWLERAAKLFPGSPAIYKLKEQLLDCEGEDGWNKLFDLIQSELYVRPDDVHVNIRLVEVYRSTKRLKDAVAHCHEAERNIALRSSLEWNSCVVQTLKEYLESLQCLESDKSDWRATNTDLLLAYANLMLLTLSTRDVQESRELLQSFDSALQSVKSLGGNDELSATFLEMKGHFYMHAGSLLLKMGQHSSNVQWRALSELAALCYLIAFQVPRPKIKLIKGEAGQNLLEMMACDRLSQSGHMLLNLSRGKQDFLKEIVETFANKSGQSALYDALFSSQSPKDTSFLGSDDIGNIDVREPELEDLTRYDVGAIRAHNGSLQHLTWLGLQWNSLPALPGIRKWLKQLFHHLPHETSRLETNAPESICILDLEVFLLGVVYTSHLQLKEKCNSHHSSYQPLCLPLPVCKQLCTERQKSWWDAVCTLIHRKAVPGNVAKLRLLVQHEINTLRAQEKHGLQPALLVHWAECLQKTGSGLNSFYDQREYMGRSVHYWKKVLPLLKIIKKKNSIPEPIDPLFKHFHSVDIQASEIVEYEEDAHITFAILDAVNGNIEDAVTAFESIKSVVSYWNLALIFHRKAEDIENDALSPEEQEECKNYLRKTRDYLIKIIDDSDSNLSVVKKLPVPLESVKEMLNSVMQE
2BFO , Knot 123 288 0.80 40 172 271 
MTAPTVPVALVTGAAKRLGRSIAEGLHAEGYAVCLHYHRSAAEANALSATLNARRPNSAITVQADLSNVATAPVSGADGSAPVTLFTRCAELVAACYTHWGRCDVLVNNASSFYPTPLLRNDEDGHEPCVGDREAMETATADLFGSNAIAPYFLIKAFAHRVAGTPAKHRGTNYSIINMVDAMTNQPLLGYTIYTMAKGALEGLTRSAALELAPLQIRVNGVGPGLSVLVDDMPPAVWEGHRSKVPLYQRDSSAAEVSDVVIFLCSSKAKYITGTCVKVDGGYSLTRA

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 \(c\).

\(|P_{f(8GUT_1)}(2) \setminus P_{f(7MNL_1)}(2)|=41\), \(|P_{f(7MNL_1)}(2) \setminus P_{f(8GUT_1)}(2)|=121\). 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 1:110010100011100001100010001001100101111111001000110010110011000000000011100001001111101110101011001010010011111111001110101111100110001101010000000100011001001001101001100001100010001110001010010101101110000000110010110111101110000111100001001000101100100001100001111100001100010001101001001100000011101000100100000000100010010000010111011001110001000111
Pair \(Z_2\) Length of longest common subsequence
8GUT_1,7MNL_1 162 4
8GUT_1,2BFO_1 174 3
7MNL_1,2BFO_1 190 4

Newick tree

 
[
	2BFO_1:94.21,
	[
		8GUT_1:81,7MNL_1:81
	]:13.21
]

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 \((0.85)(0.8)(\frac{1107 }{\log_{20} 1107}-\frac{354}{\log_{20}354})=198.\)
Status Protein1 Protein2 d d1/2
Query variables 8GUT_1 7MNL_1 251 181.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]