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Parikh vectors
3KJT_1 7WXS_1 3TJS_1 Letter Amino acid
30 12 26 G Glycine
21 2 34 P Proline
12 10 30 S Serine
20 7 26 T Threonine
9 3 15 Q Glutamine
6 2 18 M Methionine
16 3 32 F Phenylalanine
6 11 22 R Arginine
21 14 18 N Asparagine
23 7 25 D Aspartic acid
26 2 30 E Glutamic acid
3 1 12 H Histidine
31 8 51 L Leucine
45 12 18 A Alanine
0 8 7 C Cysteine
23 6 29 I Isoleucine
36 6 38 K Lycine
7 6 3 W Tryptophan
17 3 16 Y Tyrosine
20 6 37 V Valine 

3KJT_1|Chain A|Maltose-binding periplasmic protein|Escherichia coli K-12 (83333)
>7WXS_1|Chain A|Lysozyme C|Gallus gallus (9031)
>3TJS_1|Chain A|Cytochrome P450 3A4|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 \(w=f(c)\)
3KJT , Knot 152 372 0.80 38 193 343 
GAKIEEGKLVIWINGLFGYNGLAEVGKKFEKDTGIKVTVEHPDKLEEKFPQVAATGDGPDIIFYAHDRFGGYAQSGLLAEITPDKAFQDKLYPFTWDAVRYNGKLIAYPIAVYALSLIYNKDLLPNPPKTWEEIPALDKELKAKGKSALMFNLQEPYFTWPLIAADGGYAFKYENGKYDIKDVGVDNAGAKAGLTFLVDLIKNKHMNADTDYSIAEAAFNKGETAMTINGPWAWSNIDTSKVNYGVTVLPTFKGQPSKPFVGVLSAGINAASPNKELAKEFLENYLLTDEGLEAVNKDKPLGAVALKSYEEELAKDPRIAATMENAQKGEIMPNIPQMSAFWYAVRTAVINAASGRQTVDEALKDAQTRITK
7WXS , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
3TJS , Knot 203 487 0.86 40 248 467 
MAYLYGTHSHGLFKKLGIPGPTPLPFLGNILSYHKGFCMFDMECHKKYGKVWGFYDGQQPVLAITDPDMIKTVLVKECYSVFTNRRPFGPVGFMKSAISIAEDEEWKRLRSLLSPTFTSGKLKEMVPIIAQYGDVLVRNLRREAETGKPVTLKDVFGAYSMDVITSTSFGVNIDSLNNPQDPFVENTKKLLRFDFLDPFFLSITVFPFLIPILEVLNICVFPREVTNFLRKSVKRMKESRLEDTQKHRVDFLQLMIDSQNSKETESHKALSDLELVAQSIIFIFAGYETTSSVLSFIMYELATHPDVQQKLQEEIDAVLPNKAPPTYDTVLQMEYLDMVVNETLRLFPIAMRLERVCKKDVEINGMFIPKGVVVMIPSYALHRDPKYWTEPEKFLPERFSKKNKDNIDPYIYTPFGSGPRNCIGMRFALMNMKLALIRVLQNFSFKPCKETQIPLKLSLGGLLQPEKPVVLKVESRDGTVSGAHHHH

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 \(c\).

\(|P_{f(3KJT_1)}(2) \setminus P_{f(7WXS_1)}(2)|=141\), \(|P_{f(7WXS_1)}(2) \setminus P_{f(3KJT_1)}(2)|=52\). 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 1:110100101111101111001110110010000110101001001000110111010110111010001110100111101010011000101101011000101110111101101100001110110010011110001010100111101001010111111011011000010001001110011101110111011000010100000110111001001101011111001000010011011101010100111111011101101000110011000110001101100001111111000000110010111010010010111011010111011001110110100010011001000100
Pair \(Z_2\) Length of longest common subsequence
3KJT_1,7WXS_1 193 3
3KJT_1,3TJS_1 165 4
7WXS_1,3TJS_1 216 3

Newick tree

 
[
	7WXS_1:10.23,
	[
		3KJT_1:82.5,3TJS_1:82.5
	]:25.73
]

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 \((0.85)(0.8)(\frac{501 }{\log_{20} 501}-\frac{129}{\log_{20}129})=110.\)
Status Protein1 Protein2 d d1/2
Query variables 3KJT_1 7WXS_1 138 94
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]