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Parikh vectors
3TDA_1 5NGS_1 8FEH_1 Letter Amino acid
37 4 31 A Alanine
29 12 10 E Glutamic acid
60 16 24 L Leucine
16 9 14 K Lycine
32 11 6 F Phenylalanine
26 8 24 T Threonine
12 1 18 N Asparagine
22 12 12 D Aspartic acid
6 2 4 C Cysteine
19 4 0 H Histidine
37 7 16 P Proline
36 13 26 V Valine
21 11 16 Q Glutamine
13 5 8 I Isoleucine
9 5 0 M Methionine
25 6 20 S Serine
5 4 9 Y Tyrosine
35 8 14 R Arginine
33 17 16 G Glycine
6 4 0 W Tryptophan 

3TDA_1|Chains A, B, C, D|Cytochrome P450 2D6|Homo sapiens (9606)
>5NGS_1|Chains A, B|7,8-dihydro-8-oxoguanine triphosphatase|Homo sapiens (9606)
>8FEH_1|Chain A[auth D]|Minor capsid protein A1 fusion|Qubevirus durum (39803)
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 \(w=f(c)\)
3TDA , Knot 196 479 0.84 40 225 442 
MAKKTSSKGKLPPGPLPLPGLGNLLHVDFQNTPYCFDQLRRRFGDVFSLQLAWTPVVVLNGLAAVREALVTHGEDTADRPPVPITQILGFGPRSQGVFLARYGPAWREQRRFSVSTLRNLGLGKKSLEQWVTEEAACLCAAFANHSGRPFRPNGLLDKAVSNVIASLTCGRRFEYDDPRFLRLLDLAQEGLKEESGFLREVLNAVPVLLHIPALAGKVLRFQKAFLTQLDELLTEHRMTWDPAQPPRDLTEAFLAEMEKAKGNPESSFNDENLRIVVADLFSAGMVTTSTTLAWGLLLMILHPDVQRRVQQEIDDVIGQVRRPEMGDQAHMPYTTAVIHEVQRFGDIVPLGVTHMTSRDIEVQGFRIPKGTTLITNLSSVLKDEAVWEKPFRFHPEHFLDAQGHFVKPEAFLPFSAGRRACLGEPLARMELFLFFTSLLQHFSFSVPTGQPRPSHHGVFAFLVSPSPYELCAVPRHHHH
5NGS , Knot 75 159 0.79 40 122 154 
GSHMGASRLYTLVLVLQPQRVLLGMKKRGFGAGRWNGFGGKVQEGETIEDGARRELQEESGLTVDALHKVGQIVFEFVGEPELMDVHVFCTDSIQGTPVESDEMRPCWFQLDQIPFKDMWPDDSYWFPLLLQKKKFHGYFKFQGQDTILDYTLREVDTV
8FEH , Knot 68 268 0.47 34 106 133 
AKLETVTLGNIGKDGKQTLVLNPRGVNPTNGVASLSQAGAVPALEKRVTVSVSQPSRNRKNYKVQVKIQNPTACTANGSCDPSVTRQAYADVTFSFTQYSTDEERAFVRTELAALLASPLLIDAIDQLNPAYAYGGAKLETVTLGNIGKDGKQTLVLNPRGVNPTNGVASLSQAGAVPALEKRVTVSVSQPSRNRKNYKVQVKIQNPTACTANGSCDPSVTRQAYADVTFSFTQYSTDEERAFVRTELAALLASPLLIDAIDQLNPAY

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 \(c\).

\(|P_{f(3TDA_1)}(2) \setminus P_{f(5NGS_1)}(2)|=146\), \(|P_{f(5NGS_1)}(2) \setminus P_{f(3TDA_1)}(2)|=43\). 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 1:11000000101111111111110110101000100100100011011010111011111011111001110010001001111100111111000111110011110000010100100111100010011000110101111000101101011100110011101001001000010110110110011000011100110111111011111101101001110010011000010101101100100111101001010100010000101111011011110000011111111110101000100010011101001011001011000111001001101111110010000101011011010011001001100011100110101001101010110101111101100101101110101111100110010101101010100011111110101001011100000
Pair \(Z_2\) Length of longest common subsequence
3TDA_1,5NGS_1 189 4
3TDA_1,8FEH_1 185 3
5NGS_1,8FEH_1 156 4

Newick tree

 
[
	3TDA_1:98.13,
	[
		8FEH_1:78,5NGS_1:78
	]:20.13
]

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 \((0.85)(0.8)(\frac{638 }{\log_{20} 638}-\frac{159}{\log_{20}159})=137.\)
Status Protein1 Protein2 d d1/2
Query variables 3TDA_1 5NGS_1 174 115.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]