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Parikh vectors
9FFI_1 5FJG_1 6HOS_1 Letter Amino acid
13 5 10 R Arginine
0 2 4 C Cysteine
21 4 15 E Glutamic acid
23 9 3 K Lycine
13 16 19 S Serine
5 10 13 N Asparagine
21 11 18 D Aspartic acid
9 13 9 Q Glutamine
15 14 4 M Methionine
3 8 1 W Tryptophan
14 23 9 V Valine
11 16 17 F Phenylalanine
14 6 9 P Proline
17 17 23 T Threonine
11 31 12 A Alanine
31 19 21 G Glycine
11 2 10 H Histidine
7 24 12 I Isoleucine
14 31 18 L Leucine
14 14 15 Y Tyrosine 

9FFI_1|Chain A|mScarlet|synthetic construct (32630)
>5FJG_1|Chain A|CHLORIDE PUMPING RHODOPSIN|NONLABENS MARINUS S1-08 (1454201)
>6HOS_1|Chains A, B|BA75_04148T0|Komagataella pastoris (4922)
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 \(w=f(c)\)
9FFI , Knot 115 267 0.80 38 168 253 
MRGSHHHHHHGMASMTGGQQMGRDLYDDDDKDPVSKGEAVIKEFMRFKVHMEGSMNGHEFEIEGEGEGRPYEGTQTAKLKVTKGGPLPFSWDILSPQFMYGMYGSRAFTKHPADIPDYYKQSFPEGFKWERVMNFEDGGAVTVTQDTSLEDGTLIYKVKLRGTNFPPDGPVMQKKTMGWEASTERLYPEDGVLKGDIKMALRLKDGGRYLADFKTTYKAKKPVQMPGAYNVDRKLDITSHNEDYTVVEQYERSEGRHSTGGMDELYK
5FJG , Knot 125 275 0.85 40 167 266 
PNSMKNIESLFDYSAGQFEFIDHLLTMGVGVHFAALIFFLVVSQFVAPKYRIATALSCIVMVSAGLILNSQAVMWTDAYAYVDGSYQLQDLTFSNGYRYVNWMATIPCLLLQLLIVLNLKGKELFSTATWLILAAWGMIITGYVGQLYEVDDIAQLMIWGAVSTAFFVVMNWIVGTKIFKNRATMLGGTDSTITKVFWLMMFAWTLYPIAYLVPAFMNNADGVVLRQLLFTIADISSKVIYGLMITYIAIQQSAAAGYVPAQQALGRIGMDSKAA
6HOS , Knot 108 242 0.81 40 163 234 
MGSSHHHHHHSSGLVPRGSHMQESGDGDISDSAYACDIDATRYDGFNATIYEFQPGDGRLTRDPVFMSTGYLNRTQLHSITGVTDPGFSIYTPGVPTTTLYGIPNVNWENLLLELKGYFRAEVSGDYGLSLRNIDDSAILFFGKETAFQCCNENSISNEASTDYSLFTIFRQEGDETTNLDSFTYTQYLEAGKYYPVRTFFVNIERHAVFNFTMTLPDGTELTDFHNYIYQFGALDEEQCQA

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 \(c\).

\(|P_{f(9FFI_1)}(2) \setminus P_{f(5FJG_1)}(2)|=95\), \(|P_{f(5FJG_1)}(2) \setminus P_{f(9FFI_1)}(2)|=94\). 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 1:101000000011101011001100100000001100101110011010101010101001010101010100100010101001111110101101011011010011000110110000001101101001101001111010000010010110010101001110111100001110100001010011101010111010011001101000001001101111001000101000000001100000001000011100100
Pair \(Z_2\) Length of longest common subsequence
9FFI_1,5FJG_1 189 3
9FFI_1,6HOS_1 159 7
5FJG_1,6HOS_1 152 4

Newick tree

 
[
	9FFI_1:90.78,
	[
		6HOS_1:76,5FJG_1:76
	]:14.78
]

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 \((0.85)(0.8)(\frac{542 }{\log_{20} 542}-\frac{267}{\log_{20}267})=78.0\)
Status Protein1 Protein2 d d1/2
Query variables 9FFI_1 5FJG_1 103 100
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]