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Parikh vectors
6RNV_1 4CNL_1 8SXD_1 Letter Amino acid
29 19 35 A Alanine
19 0 9 P Proline
4 14 1 W Tryptophan
5 4 13 M Methionine
15 1 13 F Phenylalanine
12 5 16 T Threonine
9 28 10 Y Tyrosine
4 0 4 C Cysteine
23 5 24 E Glutamic acid
24 21 32 G Glycine
15 0 8 H Histidine
20 9 20 V Valine
20 17 16 S Serine
14 9 16 N Asparagine
10 13 16 D Aspartic acid
40 8 22 L Leucine
23 15 23 K Lycine
19 2 11 R Arginine
10 11 8 Q Glutamine
15 1 33 I Isoleucine 

6RNV_1|Chain A|Thermosynechococcus elongatus protochlorophyllide oxidoreductase (POR)|Thermosynechococcus elongatus (146786)
>4CNL_1|Chain A|PUTATIVE PNEUMOCOCCAL SURFACE PROTEIN|STREPTOCOCCUS PNEUMONIAE (1313)
>8SXD_1|Chain A|Ketol-acid reductoisomerase|Campylobacter jejuni (197)
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 \(w=f(c)\)
6RNV , Knot 141 330 0.82 40 194 311 
MSDQPRPTVIITGASSGVGLYATKALANRGWHVIMACRNLEKAEQAAKNLQIPPEAYTILHLDLSSLASVRGFVESFRALNRPLRALVCNAAVYYPLLKEPIYSVDGYEITVATNHLGHFLLINLLLEDLKNSPESDKRLVILGTVTANRKELGGKIPIPAPPDLGNLEGFEKGFKKPIAMINGKPFKSGKAYKDSKLCNMLTARELHRRFHESTGIVFNSLYPGCVADTPLFRHHFPLFQKLFPLFQKKITGGYVSQELAGERVAMVVADPEFRQSGVHWSWGNRQKEGRKAFVQELSAEASDEQKARRLWELSEKLVGLALEHHHHHH
4CNL , Knot 54 182 0.51 34 77 111 
TTSGWVKQDGAWYYFDGNGNLVKNAWQGSYYLKADGKMAQSEWIYDSSYQAWYYLKSDGSYAKNAWQGAYYLKSNGKMAQGEWVYDSSYQAWYYLKSDGSYARNAWQGNYYLKSDGKMAKGEWVYDATYQAWYYLTSDGSYAYSTWQGNYYLKSDGKMAVNEWVDGGRYYVGADGVWKEVQA
8SXD , Knot 143 330 0.83 40 192 313 
MAITVYYDKDCDLNLIKSKKVAIIGFGSQGHAHAMNLRDNGVNVTIGLREGSVSAVKAKNAGFEVMSVSEASKIADVIMILAPDEIQADIFNVEIKPNLSEGKAIAFAHGFNIHYGQIVVPKGVDVIMIAPKAPGHTVRNEFTLGGGTPCLIAIHQDESKNAKNLALSYASAIGGGRTGIIETTFKAETETDLFGEQAVLCGGLSALIQAGFETLVEAGYEPEMAYFECLHEMKLIVDLIYQGGIADMRYSISNTAEYGDYITGPKIITEETKKAMKGVLKDIQNGVFAKDFILERRAGFARMHAERKNMNDSLIEKTGRNLRAMMPWIS

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 \(c\).

\(|P_{f(6RNV_1)}(2) \setminus P_{f(4CNL_1)}(2)|=149\), \(|P_{f(4CNL_1)}(2) \setminus P_{f(6RNV_1)}(2)|=32\). 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 1:100010101110110011110100111001101111000100100110010111010011010100110101110010110011011100111001110011001010010110001101111011100100010000011111010100001110111111101101011001100111110101100101000001001101001000100001111001011011001110001111001111100010110100011100111111010100011010110000010011100101010000010011010001111110000000
Pair \(Z_2\) Length of longest common subsequence
6RNV_1,4CNL_1 181 3
6RNV_1,8SXD_1 164 4
4CNL_1,8SXD_1 179 4

Newick tree

 
[
	4CNL_1:92.51,
	[
		6RNV_1:82,8SXD_1:82
	]:10.51
]

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 \((0.85)(0.8)(\frac{512 }{\log_{20} 512}-\frac{182}{\log_{20}182})=95.9\)
Status Protein1 Protein2 d d1/2
Query variables 6RNV_1 4CNL_1 125 84.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]