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Parikh vectors
3MFE_1 8FSB_1 8SCB_1 Letter Amino acid
0 7 4 C Cysteine
8 15 10 H Histidine
15 35 16 I Isoleucine
18 60 11 L Leucine
7 35 14 P Proline
16 45 8 S Serine
19 41 23 V Valine
13 29 28 R Arginine
5 24 5 Q Glutamine
5 7 3 M Methionine
6 27 7 F Phenylalanine
11 28 11 T Threonine
9 20 6 Y Tyrosine
19 29 7 D Aspartic acid
13 28 10 E Glutamic acid
28 47 33 G Glycine
8 19 26 K Lycine
1 17 1 W Tryptophan
34 24 25 A Alanine
4 16 9 N Asparagine 

3MFE_1|Chains A[auth 2], B[auth C], C[auth E], E[auth H], F[auth J], G[auth L], H[auth N], I[auth P], J[auth R], L[auth T], M[auth X], N[auth Z]|Proteasome subunit beta|Mycobacterium tuberculosis (1773)
>8FSB_1|Chains A, B, C, D, E|5-hydroxytryptamine receptor 3A|Mus musculus (10090)
>8SCB_1|Chain A|Ribosomal protein L8|Oryctolagus cuniculus (9986)
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 \(w=f(c)\)
3MFE , Knot 108 240 0.82 40 147 227 
XTIVALKYPGGVVMAGDRRSTQGNMISGRDVRKVYITDDYTATGIAGTAAVAVEFARLYAVELEHYEKLEGVPLTFAGKINRLAIMVRGNLAAAMQGLLALPLLAGYDIHASDPQSAGRIVSFDAAGGWNIEEEGYQAVGSGSLFAKSSMKKLYSQVTDGDSGLRVAVEALYDAADDDSATGGPDLVRGIFPTAVIIDADGAVDVPESRIAELARAIIESRSGADTFGSDGGEKHHHHHH
8FSB , Knot 203 553 0.77 40 243 463 
WSHPQFEKGGGSGGGSGGGSWSHPQFEKGGGSGGGSGGGSWSHPQFEKGGGSGGGSGGGSWSHPQFEKENLYFQGATQARDTTQPALLRLSDHLLANYKKGVRPVRDWRKPTTVSIDVIMYAILNVDEKNQVLTTYIWYRQYWTDEFLQWTPEDFDNVTKLSIPTDSIWVPDILINEFVDVGKSPNIPYVYVHHRGEVQNYKPLQLVTACSLDIYNFPFDVQNCSLTFTSWLHTIQDINITLWRSPEEVRSDKSIFINQGEWELLEVFPQFKEFSIDISNSYAEMKFYVIIRRRPLFYAVSLLLPSIFLMVVDIVGFCLPPDSGERVSFKITLLLGYSVFLIIVSDTLPATAIGTPLIGVYFVVCMALLVISLAETIFIVRLVHKQDLQRPVPDWLRHLVLDRIAWILCLGEQPMAHRPPATFQANKTDDCSGSDLLPAMGNHCSHVGGPQDLEKTPRGRGSPLPPPREASLAVRGLLQELSSIRHFLEKRDEMREVARDWLRVGYVLDRLLFRIYLLAVLAYSITLVTLWSIWHYSENLYFQGTETSQVAPA
8SCB , Knot 116 257 0.83 40 162 245 
MGRVIRGQRKGAGSVFRAHVKHRKGAARLRAVDFAERHGYIKGIVKDIIHDPGRGAPLAKVVFRDPYRFKKRTELFIAAEGIHTGQFVYCGKKAQLNIGNVLPVGTMPEGTIVCCLEEKPGDRGKLARASGNYATVISHNPETKKTRVKLPSGSKKVISSANRAVVGVVAGGGRIDKPILKAGRAYHKYKAKRNCWPRVRGVAMNPVEHPFGGGNHQHIGKPSTIRRDAPAGRKVGLIAARRTGRLRGTKTVQEKEN

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 \(c\).

\(|P_{f(3MFE_1)}(2) \setminus P_{f(8FSB_1)}(2)|=44\), \(|P_{f(8FSB_1)}(2) \setminus P_{f(3MFE_1)}(2)|=140\). 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 1:001111001111111100000010110100100101000001011110111110110101101000001011110111010011111010111110111111111100101001001101101011111010001001110101110001001000100100110111011001100001011101101111011110101110110001101101110000110011001100000000
Pair \(Z_2\) Length of longest common subsequence
3MFE_1,8FSB_1 184 3
3MFE_1,8SCB_1 153 4
8FSB_1,8SCB_1 183 4

Newick tree

 
[
	8FSB_1:96.29,
	[
		3MFE_1:76.5,8SCB_1:76.5
	]:19.79
]

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 \((0.85)(0.8)(\frac{793 }{\log_{20} 793}-\frac{240}{\log_{20}240})=152.\)
Status Protein1 Protein2 d d1/2
Query variables 3MFE_1 8FSB_1 181 131.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]