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Parikh vectors
3OFJ_1 6LSS_1 5MIU_1 Letter Amino acid
22 39 46 A Alanine
5 12 9 H Histidine
4 80 34 K Lycine
6 13 9 M Methionine
15 37 23 T Threonine
3 17 12 Y Tyrosine
6 33 19 N Asparagine
7 23 19 F Phenylalanine
13 51 37 S Serine
4 8 7 W Tryptophan
14 55 46 V Valine
6 4 3 C Cysteine
15 70 40 E Glutamic acid
13 38 53 G Glycine
11 36 34 I Isoleucine
21 47 39 L Leucine
20 49 33 R Arginine
15 40 26 D Aspartic acid
8 41 18 Q Glutamine
8 38 28 P Proline 

3OFJ_1|Chain A|Nodulation protein S|Bradyrhizobium sp. (133505)
>6LSS_1|Chain A[auth 1]|Nucleolar GTP-binding protein 2|Homo sapiens (9606)
>5MIU_1|Chains A, B|Apoptosis-inducing factor 1, mitochondrial|Mus musculus (10090)
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 \(w=f(c)\)
3OFJ , Knot 103 216 0.85 40 154 208 
GIDPFTMVSVDNTYQSLERELANDDPWRLDDNPFERERHTQLLRLSLSSGAVSNGLEIGCAAGAFTEKLAPHCKRLTVIDVMPRAIGRACQRTKRWSHISWAATDILQFSTAELFDLIVVAEVLYYLEDMTQMRTAIDNMVKMLAPGGHLVFGSARDATCRRWGHVAGAETVITILTEALTEVERVQCQGQSADEDCLLARFRNPERSSIRPDGRA
6LSS , Knot 275 731 0.82 40 278 654 
MVKPKYKGRSTINPSKASTNPDRVQGAGGQNMRDRATIRRLNMYRQKERRNSRGKIIKPLQYQSTVASGTVARVEPNIKWFGNTRVIKQSSLQKFQEEMDTVMKDPYKVVMKQSKLPMSLLHDRIRPHNLKVHILDTESFETTFGPKSQRKRPNLFASDMQSLIENAEMSTESYDQGKDRDLVTEDTGVRNEAQEEIYKKGQSKRIWGELYKVIDSSDVVVQVLDARDPMGTRSPHIETYLKKEKPWKHLIFVLNKCDLVPTWATKRWVAVLSQDYPTLAFHASLTNPFGKGAFIQLLRQFGKLHTDKKQISVGFIGYPNVGKSSVINTLRSKKVCNVAPIAGETKVWQYITLMRRIFLIDCPGVVYPSEDSETDIVLKGVVQVEKIKSPEDHIGAVLERAKPEYISKTYKIDSWENAEDFLEKLAFRTGKLLKGGEPDLQTVGKMVLNDWQRGRIPFFVKPPNAEPLVAPQLLPSSSLEVVPEAAQNNPGEEVTETAGEGSESIIKEETEENSHCDANTEMQQILTRVRQNFGKINVVPQFSGDDLVPVEVSDLEEELESFSDEEEEEQEQQRDDAEESSSEPEEENVGNDTKAVIKALDEKIAKYQKFLDKAKAKKFSAVRISKGLSEKIFAKPEEQRKTLEEDVDDRAPSKKGKKRKAQREEEQEHSNKAPRALTSKERRRAVRQQRPKKVGVRYYETHNVKNRNRNKKKTNDSEGQKHKRKKFRQKQ
5MIU , Knot 214 535 0.83 40 251 504 
GAGAYAYKTIKEDQKRYNERVMGLGLSPEEKQRRAIASATEGGSVPQIRAPSHVPFLLIGGGTAAFAAARSIRARDPGARVLIVSEDPELPYMRPPLSKELWFSDDPNVTKTLQFRQWNGKERSIYFQPPSFYVSAQDLPNIENGGVAVLTGKKVVHLDVRGNMVKLNDGSQITFEKCLIATGGTPRSLSAIDRAGAEVKSRTTLFRKIGDFRALEKISREVKSITVIGEGFLGSELACALGRKSQASGIEVIQLFPEKGNMGKILPQYLSNWTMEKVKREGVKVMPNAIVQSVGVSGGRLLIKLKDGRKVETDHIVTAVGLEPNVELAKTGGLEIDSDFGGFRVNAELQARSNIWVAGDAACFYDIKLGRRRVEHHDHAVVSGRLAGENMTGAAKPYWHQSMFWSDLGPDVGYEAIGLVDSSLPTVGVFAKATAQDNPKSATEQSGTGIRSESETESEASEITIPPSAPAVPQVPVEGEDYGKGVIFYLRDKVVVGIVLWNVFNRMPIARKIIKDGEQHEDLNEVAKLFNIHED

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 \(c\).

\(|P_{f(3OFJ_1)}(2) \setminus P_{f(6LSS_1)}(2)|=30\), \(|P_{f(6LSS_1)}(2) \setminus P_{f(3OFJ_1)}(2)|=154\). 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 1:110110110100000010001100011010001100000001101010011100110110111110001110000101101110111010000001001011100110100101101111101100100100100110011011111101111010010000110111100110110011001001000100100001110100100001010101
Pair \(Z_2\) Length of longest common subsequence
3OFJ_1,6LSS_1 184 4
3OFJ_1,5MIU_1 173 4
6LSS_1,5MIU_1 123 4

Newick tree

 
[
	3OFJ_1:96.79,
	[
		5MIU_1:61.5,6LSS_1:61.5
	]:35.29
]

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 \((0.85)(0.8)(\frac{947 }{\log_{20} 947}-\frac{216}{\log_{20}216})=199.\)
Status Protein1 Protein2 d d1/2
Query variables 3OFJ_1 6LSS_1 250 160
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]