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Parikh vectors
3VKJ_1 9OIM_1 1WVQ_1 Letter Amino acid
0 2 2 C Cysteine
10 4 2 Q Glutamine
3 1 4 H Histidine
12 3 2 M Methionine
14 8 8 P Proline
4 0 1 W Tryptophan
27 5 23 V Valine
34 7 14 A Alanine
17 7 10 R Arginine
32 8 16 E Glutamic acid
29 7 13 K Lycine
16 7 7 F Phenylalanine
14 0 8 N Asparagine
26 10 12 L Leucine
25 5 4 S Serine
12 9 6 T Threonine
8 1 3 Y Tyrosine
14 10 7 D Aspartic acid
32 5 10 G Glycine
39 5 15 I Isoleucine 

3VKJ_1|Chains A, B, C, D|Isopentenyl-diphosphate delta-isomerase|Sulfolobus shibatae (2286)
>9OIM_1|Chains A, D, J|Elongin-B|Homo sapiens (9606)
>1WVQ_1|Chains A, B, C|hypothetical protein PAE2307|Pyrobaculum aerophilum (13773)
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 \(w=f(c)\)
3VKJ , Knot 157 368 0.84 38 197 345 
MPDIVNRKVEHVEIAAFENVDGLSSSTFLNDVILVHQGFPGISFSEINTKTKFFRKEISVPVMVTGMTGGRNELGRINKIIAEVAEKFGIPMGVGSQRVAIEKAEARESFAIVRKVAPTIPIIANLGMPQLVKGYGLKEFQDAIQMIEADAIAVHLNPAQEVFQPEGEPEYQIYALEKLRDISKELSVPIIVKESGNGISMETAKLLYSYGIKNFDTSGQGGTNWIAIEMIRDIRRGNWKAESAKNFLDWGVPTAASIMEVRYSVPDSFLVGSGGIRSGLDAAKAIALGADIAGMALPVLKSAIEGKESLEQFFRKIIFELKAAMMLTGSKDVDALKKTSIVILGKLKEWAEYRGINLSIYEKVRKRE
9OIM , Knot 55 104 0.81 36 88 101 
MDVFLMIRRHKTTIFTDAKESSTVFELKRIVEGILKRPPDEQRLYKDDQLLDDGKTLGECGFTSQTARPQAPATVGLAFRADDTFEALCIEPFSSPPELPDVMK
1WVQ , Knot 81 167 0.82 40 123 163 
MTDMSIKFELIDVPIPQGTNVIIGQAHFIKTVEDLYEALVTSVPGVKFGIAFCEASGKRLVRHEANDEELRNLAIDLCKKIAAGHVFVIYIRNAWPINVLNAIKNVPEVVRIFAATANPLKVIVAEVEPERRGVVGVVDGHSPLGVETEKDREERKKFLREVVKYKL

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 \(c\).

\(|P_{f(3VKJ_1)}(2) \setminus P_{f(9OIM_1)}(2)|=137\), \(|P_{f(9OIM_1)}(2) \setminus P_{f(3VKJ_1)}(2)|=28\). 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 1:11011000100101111001011000011001111001111101001000001100010111110110110001101001110110011111111000111001010001111001110111110111101101011001001101101011110101100110101010001011001001000101111100010110100101100011001000101100111101100100101010010011011110110110100011001111011100110110111111011111111100110100010011001110101111101000101100001111101001100011010100010000
Pair \(Z_2\) Length of longest common subsequence
3VKJ_1,9OIM_1 165 4
3VKJ_1,1WVQ_1 152 4
9OIM_1,1WVQ_1 131 3

Newick tree

 
[
	3VKJ_1:83.41,
	[
		1WVQ_1:65.5,9OIM_1:65.5
	]:17.91
]

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 \((0.85)(0.8)(\frac{472 }{\log_{20} 472}-\frac{104}{\log_{20}104})=110.\)
Status Protein1 Protein2 d d1/2
Query variables 3VKJ_1 9OIM_1 137 86
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]