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Parikh vectors
8SGL_1 1GBX_1 6HVR_1 Letter Amino acid
34 11 23 G Glycine
14 2 12 P Proline
28 5 19 T Threonine
27 14 9 R Arginine
26 10 6 N Asparagine
45 5 15 I Isoleucine
38 5 18 K Lycine
34 6 23 S Serine
55 14 18 A Alanine
11 2 9 F Phenylalanine
1 5 2 W Tryptophan
36 3 17 E Glutamic acid
9 8 0 C Cysteine
17 6 6 Q Glutamine
6 1 3 H Histidine
55 9 26 L Leucine
14 2 4 M Methionine
8 6 10 Y Tyrosine
45 8 16 V Valine
33 8 14 D Aspartic acid 

8SGL_1|Chains A, I[auth a]|T-complex protein 1 subunit alpha|Homo sapiens (9606)
>1GBX_1|Chain A|LYSOZYME|Homo sapiens (9606)
>6HVR_1|Chains A, O|Proteasome subunit alpha type-2|Saccharomyces cerevisiae S288C (559292)
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 \(w=f(c)\)
8SGL , Knot 214 536 0.83 40 229 495 
EGPLSVFGDRSTGETIRSQNVMAAASIANIVKSSLGPVGLDKMLVDDIGDVTITNDGATILKLLEVEHPAAKVLCELADLQDKEVGDGTTSVVIIAAELLKNADELVKQKIHPTSVISGYRLACKEAVRYINENLIVNTDELGRDCLINAAKTSMSSKIIGINGDFFANMVVDAVLAIKYTDIRGQPRYPVNSVNILKAHGRSQMESMLISGYALNCVVGSQGMPKRIVNAKIACLDFSLQKTKMKLGVQVVITDPEKLDQIRQRESDITKERIQKILATGANVILTTGGIDDMCLKYFVEAGAMAVRRVLKRDLKRIAKASGATILSTLANLEGEETFEAAMLGQAEEVVQERICDDELILIKNTKARTSASIILRGANDFMCDEMERSLHDALCVVKRVLESKSVVPGGGAVEAALSIYLENYATSMGSREQLAIAEFARSLLVIPNTLAVNAAQDSTDLVAKLRAFHNEAQVNPERKNLKWIGLDLSNGKPRDNKQAGVFEPTIVKVKSLKFATEAAITILRIDDLIKLHPES
1GBX , Knot 68 130 0.84 40 109 128 
KVFERCELARTLKRLGMDGYRGISLANWMCLAKWESGYNTRATNYNAGDRSTDYGIFQINSRYWCNDGKTPGAVNACHLSCSALLQDNIADAVACAKRVVRDPQGIRAWLAWRNRCQNRDVRQYVQGCGV
6HVR , Knot 111 250 0.81 38 159 239 
MTDRYSFSLTTFSPSGKLGQIDYALTAVKQGVTSLGIKATNGVVIATEKKSSSPLAMSETLSKVSLLTPDIGAVYSGMGPDYRVLVDKSRKVAHTSYKRIYGEYPPTKLLVSEVAKIMQEATQSGGVRPFGVSLLIAGHDEFNGFSLYQVDPSGSYFPWKATAIGKGSVAAKTFLEKRWNDELELEDAIHIALLTLKESVEGEFNGDTIELAIIGDENPDLLGYTGIPTDKGPRFRKLTSQEINDRLEAL

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 \(c\).

\(|P_{f(8SGL_1)}(2) \setminus P_{f(1GBX_1)}(2)|=151\), \(|P_{f(1GBX_1)}(2) \setminus P_{f(8SGL_1)}(2)|=31\). 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 1:01110111000010010000111110110110001111110011100110101000110110110100111011001101000011010001111110110010011000101001101001100011001000111000011000110110001000111101011101110111110000101010011001011010100010011101011001110011100110101101010100001011101110010010010000001000010011101101110011100101001101111110011000100110101101100110101000101111101001100010000111100001000101110110011000100010011011001100001111111101110101000100110000111101100111110011101100000111010110001010100001011110100101000001111010110100101100111011010011010100
Pair \(Z_2\) Length of longest common subsequence
8SGL_1,1GBX_1 182 5
8SGL_1,6HVR_1 158 4
1GBX_1,6HVR_1 176 3

Newick tree

 
[
	1GBX_1:92.75,
	[
		8SGL_1:79,6HVR_1:79
	]:13.75
]

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 \((0.85)(0.8)(\frac{666 }{\log_{20} 666}-\frac{130}{\log_{20}130})=154.\)
Status Protein1 Protein2 d d1/2
Query variables 8SGL_1 1GBX_1 189 115.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]