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Parikh vectors
8JNP_1 1FIB_1 8BRI_1 Letter Amino acid
46 18 28 A Alanine
17 14 20 I Isoleucine
31 7 6 P Proline
8 13 3 Y Tyrosine
31 8 20 V Valine
38 6 10 R Arginine
20 22 20 D Aspartic acid
35 10 11 E Glutamic acid
12 16 15 F Phenylalanine
4 4 0 C Cysteine
27 33 26 G Glycine
18 18 12 T Threonine
11 14 7 S Serine
5 10 0 W Tryptophan
6 15 8 N Asparagine
12 14 6 Q Glutamine
11 8 2 H Histidine
48 13 29 L Leucine
3 21 13 K Lycine
13 5 17 M Methionine 

8JNP_1|Chain A|Cytochrome P450 CftA|Streptomyces torulosus (68276)
>1FIB_1|Chain A|GAMMA-FIBRINOGEN CARBOXYL TERMINAL FRAGMENT|Homo sapiens (9606)
>8BRI_1|Chains A, B, C, D, E|Chemotaxis protein PomA|Vibrio alginolyticus (663)
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 \(w=f(c)\)
8JNP , Knot 161 396 0.81 40 197 372 
MPDQHPPLPYPFAPRGLDLDPAYAELRERAPARIRMPYGDDAWLVTRYEDVRTVLADPRFSLAASMGRDQPRMRPVARTGAGLFSTEPPEHTRLRSLVARQFSARRVEPLRARAGQLADELIDGMVVAGQPADLVEDFAIPMPTTIICEVLGIPAKDHRMLWHWAETVLSAITPQEVLATEGRAFVEYMVGVLELRGREPGDDLLTALVRACREEGLISEEELLSIACDLLIAGFVSTTNQIGNFFHQLLVHPEELTRLRERPELIPKAVEELMRYVPLLTGFNLARYATADVELGGITIRAGEAVMIATAAANRDPEVFQEPERLVMDRGANPHIGFGHGVHYCVGAHLARLELQVAIERVLHRLPGMRLAVPESELSWKQDAMVNGLQALPVAW
1FIB , Knot 123 269 0.85 40 190 256 
VQIHDITGKDCQDIANKGAKQSGLYFIKPLKANQQFLVYCEIDGSGNGWTVFQKRLDGSVDFKKNWIQYKEGFGHLSPTGTTEFWLGNEKIHLISTQSAIPYALRVELEDWNGRTSTADYAMFKVGPEADKYRLTYAYFAGGDAGDAFDGFDFGDDPSDKFFTSHNGMQFSTWDNDNDKFEGNCAEQDGSGWWMNKCHAGHLNGVYYQGGTYSKASTPNGYDNGIIWATWKTRWYSMKKTTMKIIPFNRLTIGEGQQHHLGGAKQAGDV
8BRI , Knot 110 253 0.80 36 148 244 
MDLATLLGLIGGFAFVIMAMVLGGSIGMFVDVTSILIVVGGSIFVVLMKFTMGQFFGATKIAGKAFMFKADEPEDLIAKIVEMADAARKGGFLALEEMEINNTFMQKGIDLLVDGHDADVVRAALKKDIALTDERHTQGTGVFRAFGDVAPAMGMIGTLVGLVAMLSNMDDPKAIGPAMAVALLTTLYGAILSNMVFFPIADKLSLRRDQETLNRRLIMDGVLAIQDGQNPRVIDSYLKNYLNEGKRALEIDE

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 \(c\).

\(|P_{f(8JNP_1)}(2) \setminus P_{f(1FIB_1)}(2)|=101\), \(|P_{f(1FIB_1)}(2) \setminus P_{f(8JNP_1)}(2)|=94\). 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 1:110001111011110110101101010001110101101001111000001001110101011101100010101110011111000110000100111001010010110101101100110111111011011001111110011001111110000111011001101101001110010111001111101010011001101110100001110000110110011111110000011011001110100100100010111011001100111101101100101010111101011011111011100010110010011100110101111011000111011010101110011001111011110001010001110110111111
Pair \(Z_2\) Length of longest common subsequence
8JNP_1,1FIB_1 195 4
8JNP_1,8BRI_1 169 3
1FIB_1,8BRI_1 174 3

Newick tree

 
[
	1FIB_1:94.88,
	[
		8JNP_1:84.5,8BRI_1:84.5
	]:10.38
]

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 \((0.85)(0.8)(\frac{665 }{\log_{20} 665}-\frac{269}{\log_{20}269})=110.\)
Status Protein1 Protein2 d d1/2
Query variables 8JNP_1 1FIB_1 139 118
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]