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Parikh vectors
7PFR_1 4YET_1 4RAS_1 Letter Amino acid
16 12 46 E Glutamic acid
21 8 17 H Histidine
2 4 14 C Cysteine
41 20 65 G Glycine
14 11 29 I Isoleucine
11 8 19 Y Tyrosine
9 4 26 M Methionine
5 9 9 W Tryptophan
20 13 87 A Alanine
10 15 17 N Asparagine
18 9 52 D Aspartic acid
6 3 18 Q Glutamine
12 19 55 L Leucine
8 13 27 K Lycine
21 6 41 V Valine
15 4 63 R Arginine
7 11 21 F Phenylalanine
16 11 45 P Proline
10 14 29 S Serine
21 9 24 T Threonine 

7PFR_1|Chains A, B, C, D, E, F|Two-domain laccase|Streptomyces griseoflavus (35619)
>4YET_1|Chains A, B|Superoxide dismutase|Babesia bovis (5865)
>4RAS_1|Chains A, B, C|Oxidoreductase, NAD-binding/iron-sulfur cluster-binding protein|Nitratireductor pacificus pht-3B (391937)
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 \(w=f(c)\)
7PFR , Knot 125 283 0.83 40 182 269 
AGAAPAGGEVRRVTMYAERLAGGQMGYGLEKGKASIPGPLIELNEGDTLHVEFENTMDVPVSLHVHGLDYEISSDGTKQNKSHVEPGGTRTYTWRTHEPGRRADGTWRAGSAGYWHYHDHVVGTEHGTGGIRNGLYGPVIVRRKGDVLPDATHTIVFNDATINNRPAHTGPNFEATVGDRVEIVMITHGEYYHTFHMHGHRWADNRTGMLTGPDDPSQVIDNKICGPADSFGFQIIAGEGVGAGAWMYHCHVQSHSDMGMVGLFLVKKPDGTIPGYDPQEHAH
4YET , Knot 96 203 0.83 40 156 197 
GPGSMAFKLPALPYGMRELIPHISEETLSFHYGKHHAGYVNKLNSLIKGTPMESCTIEELILGQTGAVFNNAAQIWNHTFYWNSMGPNCGGEPTGPIRKKIEEKFGSFSAFKTDFSNLLAGHFGSGWGWLVLKDDGTADIVQTHDAGSPLKENLGRPLLCCDVWEHAYYIDYKNDRLSYINSWWNLVNWDFANKNLEAPFKWS
4RAS , Knot 268 704 0.83 40 276 628 
MRLYSNRDRPNHLGPLALERLARVDDVVAQPARQPEDGFAASEDSLLGDVEEYARLFTRFLDGPVAPLGDAIPDDPARRAENLKASAYFLDASMVGICRLDPDDRAGDCDPSHTHALVFAVQFGREPEAGEAGAEWIRGTNAARTDMRCAEIAAILSGYVRWMGFPARGHFSGDAQVDLARLAVRAGLARVVDGVLVAPFLRRGFRLGVVTTGYALAADRPLAPEGDLGETAPEVMLGIDGTRPGWEDAEEEKRPLHMGRYPMETIRRVDEPTTLVVRQEIQRVAKRGDFFKRAEAGDLGEKAKQEKKRFPMKHPLALGMQPLIQNMVPLQGTREKLAPTGKGGDLSDPGRNAEAIKALGYYLGADFVGICRAEPWMYYASDEVEGKPIEAYHDYAVVMLIDQGYETMEGASGDDWISASQSMRAYMRGAEIAGVMAAHCRRMGYSARSHSNAHSEVIHNPAILMAGLGEVSRIGDTLLNPFIGPRSKSIVFTTDLPMSVDRPIDFGLQDFCNQCRKCARECPCNAISFGDKVMFNGYEIWKADVEKCTKYRVTQMKGSACGRCMKMCPWNREDTVEGRRLAELSIKVPEARAAIIAMDDALQNGKRNLIKRWWFDLEVIDGVAGAPRMGTNERDLSPDRGDKIGANQKLAMYPPRLQPPPGTTLDAVLPVDRSGGLAEYAAAETPAAARARLKSSAGHHHHHH

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 \(c\).

\(|P_{f(7PFR_1)}(2) \setminus P_{f(4YET_1)}(2)|=104\), \(|P_{f(4YET_1)}(2) \setminus P_{f(7PFR_1)}(2)|=78\). 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 1:1111111101001010100111101101100101011111101001001010100010111010101100010001000000010111000001000011001010101101101000001110001011100110111110001011101000111001010001100110101011001011110010000010101001100001110110010011000101110011101111011111111000010000011111111100101011100100010
Pair \(Z_2\) Length of longest common subsequence
7PFR_1,4YET_1 182 3
7PFR_1,4RAS_1 176 4
4YET_1,4RAS_1 204 3

Newick tree

 
[
	4YET_1:99.37,
	[
		7PFR_1:88,4RAS_1:88
	]:11.37
]

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 \((0.85)(0.8)(\frac{486 }{\log_{20} 486}-\frac{203}{\log_{20}203})=82.2\)
Status Protein1 Protein2 d d1/2
Query variables 7PFR_1 4YET_1 105 90
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]