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Parikh vectors
8QEH_1 1GEZ_1 2FKD_1 Letter Amino acid
21 13 6 A Alanine
24 14 5 R Arginine
25 8 9 D Aspartic acid
18 2 5 F Phenylalanine
18 7 9 S Serine
8 2 3 P Proline
3 5 1 W Tryptophan
17 6 2 Y Tyrosine
4 8 1 C Cysteine
15 6 3 Q Glutamine
30 3 8 E Glutamic acid
14 11 11 G Glycine
26 5 9 I Isoleucine
23 5 10 K Lycine
14 10 1 N Asparagine
5 1 1 H Histidine
35 8 11 L Leucine
9 2 1 M Methionine
21 5 5 T Threonine
22 9 9 V Valine 

8QEH_1|Chain A|Guanine nucleotide-binding protein subunit alpha-11|Homo sapiens (9606)
>1GEZ_1|Chain A|LYSOZYME|Homo sapiens (9606)
>2FKD_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N|Repressor protein CI|Enterobacteria phage 186 (29252)
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 \(w=f(c)\)
8QEH , Knot 154 352 0.85 40 205 339 
GCTLSAEDKAAVERSKMIDRNLREDGEKARRELKLLLLGTGESGKSTFIKQMRIIHGAGYSEEDKRGFTKLVYQNIFTAMQAMIRAMETLKILYKYEQNKANALLIREVDVEKVTTFEHQYVSAIKTLWEDPGIQECYDRRREYQLSDSAKYYLTDVDRIATLGYLPTQQDVLRVRVPTTGIIEYPFDLENIIFRMVDVGGQRSERRKWIHCFENVTSIMFLVALSEYDQVLVESDNENRMEESKALFRTIITYPWFQNSSVILFLNKKDLLEDKILYSHLVDYFPEFDGPQRDAQAAREFILKMFVDLNPDSDKIIYSHFTCATDTENIRFVFAAVKDTILQLNLKEYNLV
1GEZ , Knot 68 130 0.84 40 108 128 
KVFERCELSRTLKRLGMDGYRGISLANWMCLAKWESGYNTRATNYNAGDRSTDYGIFQINSRYWCNDGKTPGAVNACHLSCSALLQDNIADAVACAKRVVRDPQGIRAWVAWRNRCQNRDVRQYVQGCGV
2FKD , Knot 56 110 0.79 40 84 106 
SDAKRIEGFTLSEEILKSDKQLSVDAQFFTKPLTDGMAIRSEGKIYFVDKQASLSDGLWLVDIEGAISIRELTKLPGRKLHVAGGKVPFECGIDDIKTLGRVVGVYSEVN

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 \(c\).

\(|P_{f(8QEH_1)}(2) \setminus P_{f(1GEZ_1)}(2)|=141\), \(|P_{f(1GEZ_1)}(2) \setminus P_{f(8QEH_1)}(2)|=44\). 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 1:1001010001110000110001000100100010111110100100011001011011100000001100110001101101110110010110000000101111001010010010000101100110011100000000000100010001001001101101100001101011001110011010011101101110000000110010010011111110000011100000001000011100110011100001111100001100011000110011010110001011001110111010100001100010010000010111111000110101000011
Pair \(Z_2\) Length of longest common subsequence
8QEH_1,1GEZ_1 185 3
8QEH_1,2FKD_1 165 4
1GEZ_1,2FKD_1 134 3

Newick tree

 
[
	8QEH_1:93.51,
	[
		2FKD_1:67,1GEZ_1:67
	]:26.51
]

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 \((0.85)(0.8)(\frac{482 }{\log_{20} 482}-\frac{130}{\log_{20}130})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 8QEH_1 1GEZ_1 134 91
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]