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Parikh vectors
1QNO_1 1NFY_1 8WHC_1 Letter Amino acid
28 15 7 A Alanine
8 9 6 C Cysteine
20 9 8 Q Glutamine
15 4 6 W Tryptophan
5 15 10 R Arginine
29 14 13 S Serine
13 14 12 I Isoleucine
18 15 14 L Leucine
10 20 11 K Lycine
16 12 8 F Phenylalanine
24 17 17 V Valine
10 9 9 P Proline
33 21 13 T Threonine
28 7 8 N Asparagine
18 11 16 D Aspartic acid
8 20 9 E Glutamic acid
34 23 19 G Glycine
7 6 1 H Histidine
2 5 0 M Methionine
18 8 7 Y Tyrosine 

1QNO_1|Chain A|ENDO-1,4-B-D-MANNANASE|TRICHODERMA REESEI (51453)
>1NFY_1|Chain A|Coagulation factor XA, heavy chain|Homo sapiens (9606)
>8WHC_1|Chains A, B, C, D|21 kDa seed protein-like|Durio zibethinus (66656)
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 \(w=f(c)\)
1QNO , Knot 144 344 0.81 40 198 337 
ASSFVTISGTQFNIDGKVGYFAGTNCYWCSFLTNHADVDSTFSHISSSGLKVVRVWGFNDVNTQPSPGQIWFQKLSATGSTINTGADGLQTLDYVVQSAEQHNLKLIIPFVNNWSDYGGINAYVNAFGGNATTWYTNTAAQTQYRKYVQAVVSRYANSTAIFAWELGNEPRCNGCSTDVIVQWATSVSQYVKSLDSNHLVTLGDEGLGLSTGDGAYPYTYGEGTDFAKNVQIKSLDFGTFHLYPDSWGTNYTWGNGWIQTHAAACLAAGKPCVFEEYGAQQNPCTNEAPWQTTSLTTRGMGGDMFWQWGDTFANGAQSNSDPYTVWYNSSNWQCLVKNHVDAIN
1NFY , Knot 116 254 0.84 40 181 246 
IVGGQECKDGECPWQALLINEENEGFCGGTILSEFYILTAAHCLYQAKRFKVRVGDRNTEQEEGGEAVHEVEVVIKHNRFTKETYDFDIAVLRLKTPITFRMNVAPACLPERDWAESTLMTQKTGIVSGFGRTHEKGRQSTRLKMLEVPYVDRNSCKLSSSFIITQNMFCAGYDTKQEDACQGDSGGPHVTRFKDTYFVTGIVSWGEGCARKGKYGIYTKVTAFLKWIDRSMKTRGLPKAKSHAPEVITSSPLK
8WHC , Knot 92 194 0.83 38 143 189 
KNEPVLDIDGEELRAGEQYYVVSAIWGPGGGGLALGRLTDQKCPEIVVQRRSDSDYGTPVIFHNLDTKDDIVRLSTDFNVEFVPIRDRLCLTSTVWKIDDYDTSTGQWWVTTNGVIGNPGPQTLQSWFKIEKSGNLGYKFNFCPSVCESCVTLCSDIGRYGDDGQIRLALAERGWPFVFKKASSTIKQVVYAKN

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 \(c\).

\(|P_{f(1QNO_1)}(2) \setminus P_{f(1NFY_1)}(2)|=103\), \(|P_{f(1NFY_1)}(2) \setminus P_{f(1QNO_1)}(2)|=86\). 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 1:10011010100101010110111000010011000101000100100011011011110010001011011100101010010011011001001100100001011111100100011101010111101001000011000000010111000100011111011001000100001110110010001001000011011001111001011010001010011001010010110101010011000011011100011101111010110001100010000111000010001111011101100110110000010011000001001100010110
Pair \(Z_2\) Length of longest common subsequence
1QNO_1,1NFY_1 189 3
1QNO_1,8WHC_1 167 4
1NFY_1,8WHC_1 176 3

Newick tree

 
[
	1NFY_1:93.76,
	[
		1QNO_1:83.5,8WHC_1:83.5
	]:10.26
]

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 \((0.85)(0.8)(\frac{598 }{\log_{20} 598}-\frac{254}{\log_{20}254})=97.0\)
Status Protein1 Protein2 d d1/2
Query variables 1QNO_1 1NFY_1 124 109.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]