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Parikh vectors
3FNH_1 7HWY_1 8BOS_1 Letter Amino acid
5 6 3 H Histidine
16 14 8 S Serine
8 5 4 N Asparagine
1 6 3 C Cysteine
20 5 11 I Isoleucine
27 12 11 L Leucine
9 2 8 K Lycine
13 4 3 P Proline
36 10 11 A Alanine
13 9 11 R Arginine
4 3 0 W Tryptophan
6 8 9 Y Tyrosine
15 9 14 D Aspartic acid
11 7 13 E Glutamic acid
10 2 4 M Methionine
7 3 5 F Phenylalanine
13 6 11 T Threonine
17 14 15 V Valine
10 5 11 Q Glutamine
28 14 11 G Glycine 

3FNH_1|Chain A|Enoyl-[acyl-carrier-protein] reductase [NADH]|Mycobacterium tuberculosis (1773)
>7HWY_1|Chains A, B[auth C]|Protease 2A|Coxsackievirus A16 (31704)
>8BOS_1|Chain A[auth R]|GTPase HRas|Homo sapiens (9606)
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 \(w=f(c)\)
3FNH , Knot 115 269 0.79 40 167 253 
MTGLLDGKRILVSGIITDSSIAFHIARVAQEQGAQLVLTGFDRLRLIQRITDRLPAKAPLLELDVQNEEHLASLAGRVTEAIGAGNKLDGVVHSIGFMPQTGMGINPFFDAPYADVSKGIHISAYSYASMAKALLPIMNPGGSIVGMDFDPSRAMPAYNWMTVAKSALESVNRFVAREAGKYGVRSNLVAAGPIRTLAMSAIVGGALGEEAGAQIQLLEEGWDQRAPIGWNMKDATPVAKTVCALLSDWLPATTGDIIYADGGAHTQLL
7HWY , Knot 70 144 0.80 40 112 139 
SGAIYVGNYRVVNRHLATHNDWANLVWEDSSRDLLVSSTTAQGCDTIARCDCQTGVYYCSSRRKHYPVSFSKPSLIFVEASEYYPARYQSHLMLAVGHSEPGDCGGILRCQHGVVGIVSTGGNGLVGFADVRDLLWLDEEAMEQ
8BOS , Knot 79 166 0.81 38 130 163 
MTEYKLVVVGAGGVGKSALTIQLIQNHFVDEYDPTIEDSYRKQVVIDGETCLLDILDTAGQEEYSAMRDQYMRTGEGFLCVFAINNTKSFEDIHQYREQIKRVKDSDDVPMVLVGNKCDLAARTVESRQAQDLARSYGIPYIETSAKTRQGVEDAFYTLVREIRQH

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 \(c\).

\(|P_{f(3FNH_1)}(2) \setminus P_{f(7HWY_1)}(2)|=111\), \(|P_{f(7HWY_1)}(2) \setminus P_{f(3FNH_1)}(2)|=56\). 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 1:10111010011101110000111011011000110111011001011001000111011110101000001101110100111110010111001111100111101110110101001101010001011011111101110111101010011110011011001100100111001100110001111111001110111111110011101011001100011111010010111001011100111100101101011100011
Pair \(Z_2\) Length of longest common subsequence
3FNH_1,7HWY_1 167 3
3FNH_1,8BOS_1 155 4
7HWY_1,8BOS_1 156 3

Newick tree

 
[
	7HWY_1:81.86,
	[
		3FNH_1:77.5,8BOS_1:77.5
	]:4.36
]

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 \((0.85)(0.8)(\frac{413 }{\log_{20} 413}-\frac{144}{\log_{20}144})=80.6\)
Status Protein1 Protein2 d d1/2
Query variables 3FNH_1 7HWY_1 101 76
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]