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Parikh vectors
9FYB_1 6BZH_1 1JDV_1 Letter Amino acid
9 7 14 N Asparagine
14 14 16 K Lycine
7 13 16 S Serine
2 5 3 W Tryptophan
15 4 23 V Valine
1 5 2 C Cysteine
19 25 25 L Leucine
7 4 4 M Methionine
7 15 19 A Alanine
14 11 8 D Aspartic acid
11 13 3 Q Glutamine
14 7 22 G Glycine
4 2 2 H Histidine
13 7 11 Y Tyrosine
9 10 8 R Arginine
16 24 16 E Glutamic acid
16 12 12 I Isoleucine
8 7 9 F Phenylalanine
14 5 6 P Proline
14 4 17 T Threonine 

9FYB_1|Chains A, B|Probable nicotinate-nucleotide adenylyltransferase|Enterococcus faecium (1352)
>6BZH_1|Chains A, B, C, D, E|Probable ATP-dependent RNA helicase DDX58|Mus musculus (10090)
>1JDV_1|Chains A, B, C, D, E, F|5'-METHYLTHIOADENOSINE PHOSPHORYLASE|Sulfolobus solfataricus (2287)
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 \(w=f(c)\)
9FYB , Knot 102 214 0.85 40 155 209 
NTQAKTFVRSQLFPEEMPQFLEKKKQVGILGGTFNPVHLAHLVMAEQAGRNLGLDRVFLMPSYQPPHVDEKQTIDAKHRLNMLELAVEDNPFLQIETIELARGGKSYTYDTMKELTQNNPDTDYYFIIGGDMVEYLPKWYKIDELTSMVNFVGIRRPGYTTDTPYPVIWVDVPEIDISSTKIRQKIKEGCSIRYLVPDKVIDYIQNEGLYEYGL
6BZH , Knot 91 194 0.82 40 137 185 
GSSARQTAEQRQNLQAFRDYIKKILDPTYILSYMSSWLEDEEVQYIQAEKNNKGPMEAASLFLQYLLKLQSEGWFQAFLDALYHAGYCGLCEAIESWDFQKIEKLEEHRLLLRRLEPEFKATVDPNDILSELSECLINQECEEIRQIRDTKGRMAGAEKMAECLIRSDKENWPKVLQLALEKDNSKFSELWIVD
1JDV , Knot 108 236 0.83 40 160 226 
MNPVHILAKKGEVAERVLVVGDPGRARLLSTLLQNPKLTNENRGFLVYTGKYNGETVSIATHGIGGPSIAIVLEELAMLGANVFIRYGTTGALVPYINLGEYIIVTGASYNQGGLFYQYLRDNACVASTPDFELTNKLVTSFSKRNLKYYVGNVFSSDAFYAEDEEFVKKWSSRGNIAVEMECATLFTLSKVKGWKSATVLVVSDNLAKGGIWITKEELEKSVMDGAKAVLDTLTS

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 \(c\).

\(|P_{f(9FYB_1)}(2) \setminus P_{f(6BZH_1)}(2)|=96\), \(|P_{f(6BZH_1)}(2) \setminus P_{f(9FYB_1)}(2)|=78\). 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 1:0001001100011100110110000011111101011011011110011001110011111000110100000101000101101110001110100101101100000001001000010000011111011001101001001001101111001100000101111101101010000100010010010011100110010001100011
Pair \(Z_2\) Length of longest common subsequence
9FYB_1,6BZH_1 174 3
9FYB_1,1JDV_1 157 4
6BZH_1,1JDV_1 163 4

Newick tree

 
[
	6BZH_1:86.13,
	[
		9FYB_1:78.5,1JDV_1:78.5
	]:7.63
]

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 \((0.85)(0.8)(\frac{408 }{\log_{20} 408}-\frac{194}{\log_{20}194})=63.2\)
Status Protein1 Protein2 d d1/2
Query variables 9FYB_1 6BZH_1 85 79
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]