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Parikh vectors
6ZKV_1 4QJF_1 4EJF_1 Letter Amino acid
42 8 18 A Alanine
13 6 10 N Asparagine
13 4 7 Q Glutamine
9 2 17 H Histidine
23 8 9 P Proline
9 2 2 W Tryptophan
31 23 18 V Valine
22 11 19 D Aspartic acid
12 0 9 C Cysteine
27 16 13 I Isoleucine
33 11 25 L Leucine
25 17 20 K Lycine
9 9 6 M Methionine
23 5 14 T Threonine
10 7 10 Y Tyrosine
34 13 15 R Arginine
54 15 16 G Glycine
25 5 16 S Serine
32 22 17 E Glutamic acid
18 6 18 F Phenylalanine 

6ZKV_1|Chain A[auth 1]|NADH dehydrogenase [ubiquinone] flavoprotein 1, mitochondrial|Ovis aries (9940)
>4QJF_1|Chain A|DNA-directed RNA polymerase, subunit E'|Thermococcus kodakarensis (69014)
>4EJF_1|Chains A, B, C, D|Caspase-6|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 \(w=f(c)\)
6ZKV , Knot 187 464 0.82 40 240 435 
MLAARRLLGGSLPARVSVRFSGDTTAPKKTSFGSLKDEDRIFTNLYGRHDWRLKGAQSRGDWYKTKEILLKGPDWILGEVKTSGLRGRGGAGFPTGLKWSFMNKPSDGRPKYLVVNADEGEPGTCKDREIIRHDPHKLVEGCLVGGRAMGARAAYIYIRGEFYNEASNLQVAIREAYEAGLIGKNACGSGYDFDVFVVRGAGAYICGEETALIESIEGKQGKPRLKPPFPADVGVFGCPTTVANVETVAVSPTICRRGGAWFASFGRERNSGTKLFNISGHVNNPCTVEEEMSVPLKELIEKHAGGVTGGWDNLLAVIPGGSSTPLIPKSVCETVLMDFDALIQAQTGLGTAAVIVMDRSTDIVKAIARLIEFYKHESCGQCTPCREGVDWMNKVMARFVRGDARPAEIDSLWEISKQIEGHTICALGDGAAWPVQGLIRHFRPELEERMQRFAQQHQARQAAS
4QJF , Knot 88 190 0.81 38 127 184 
MYKLLKVKDVVRIPPRMFTMDPKEAAKIVLRETYEGIYDRDEGVVLAILDVEEISEGVIVPGDGATYHEAIFNVLVWEPRNQEVVEGEVVEMMPYGAFIRIGPMDGLVHISQLMDDYVVFDEKNRQFIGKETNRVLKLGDYVRARIIGVSVKSRVIRENKINMTMRQPGLGKFEWIEKEKKKAKEESKGE
4EJF , Knot 127 279 0.85 40 186 270 
MGSAFYKREMFDPAEKYKMDHRRRGIALIFNHERFFWHLTLPERRGTCADRDNLTRRFSDLGFEVKCFNDLKAEELLLKIHEVSTVSHADADCFVCVFLSHGEGNHIYAYDAKIEIQTLTGLFKGDKCHSLVGKPKIFIIQAARGNQHDVPVIPLDVVDNQTEKLDTNITEVDAASVYTLPAGADFLMCYSVAEGYYSHRETVNGSWYIQDLCEMLGKYGSSLEFTELLTLVNRKVSQRRVDFCKDPSAIGKKQVPCFASMLTKKLHFFPKSNHHHHHH

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 \(c\).

\(|P_{f(6ZKV_1)}(2) \setminus P_{f(4QJF_1)}(2)|=152\), \(|P_{f(4QJF_1)}(2) \setminus P_{f(6ZKV_1)}(2)|=39\). 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 1:11110011110111010101010001100001101000001100101000101011000101000001110110111101000110101111110110101100100101001110100101100000011000100110101111011110110101010100010010111001001111100101010010111101111010100011100101001010101111101111101001101001110101000111111011000001001101010100100100010111001100011110111001111111100011110010001110101110100111011111100000110111011010000001000100011011001110110101011010011010001010010111011111101110010101000100110000100110
Pair \(Z_2\) Length of longest common subsequence
6ZKV_1,4QJF_1 191 4
6ZKV_1,4EJF_1 174 3
4QJF_1,4EJF_1 177 3

Newick tree

 
[
	4QJF_1:93.69,
	[
		6ZKV_1:87,4EJF_1:87
	]:6.69
]

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 \((0.85)(0.8)(\frac{654 }{\log_{20} 654}-\frac{190}{\log_{20}190})=131.\)
Status Protein1 Protein2 d d1/2
Query variables 6ZKV_1 4QJF_1 167 116.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]