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Parikh vectors
6CGZ_1 7KLW_1 9NOT_1 Letter Amino acid
8 6 44 Q Glutamine
22 6 35 E Glutamic acid
18 13 123 L Leucine
16 8 18 K Lycine
10 0 19 M Methionine
11 16 67 V Valine
16 17 24 N Asparagine
12 14 44 F Phenylalanine
14 10 34 T Threonine
10 15 21 Y Tyrosine
12 9 49 R Arginine
3 8 32 C Cysteine
21 15 65 G Glycine
10 1 22 H Histidine
25 7 16 I Isoleucine
15 15 63 S Serine
8 2 18 W Tryptophan
22 12 82 A Alanine
15 9 36 D Aspartic acid
14 11 47 P Proline 

6CGZ_1|Chains A, B, C|Beta-lactamase|Alicyclobacillus acidoterrestris (strain ATCC 49025 / DSM 3922 / CIP 106132 / NCIMB 13137 / GD3B) (1356854)
>7KLW_1|Chain A|Spike protein S1|Severe acute respiratory syndrome coronavirus 2 (2697049)
>9NOT_1|Chain A[auth B]|Taste receptor type 1 member 3|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 \(w=f(c)\)
6CGZ , Knot 129 282 0.86 40 190 274 
MTNIAKAQPKLYVMDNGRMRMDKNWMIAMHNPATIANPNAPTEFIEFPIYTVLIDHPEGKILFDTSCNPDSMGAQGRWGEATQSMFPWTASEECYLHNRLEQLKVRPEDIKFVIASHLHLDHAGCLEMFTNATIIVHEDEFSGALQTYARNQTEGAYIWGDIDAWIKNNLNWRTIKRDEDNIVLAEGIKILNFGSGHAWGMLGLHVQLPEKGGIILASDAVYSAESYGPPIKPPGIIYDSLGFVRSVEKIKRIAKETNSEVWFGHDSEQFKRFRKSTEGYYE
7KLW , Knot 91 194 0.82 38 138 187 
NLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGP
9NOT , Knot 315 859 0.82 40 299 759 
MKTIIALSYIFCLVFADYKDDDDKAAAAPLCLSQQLRMKGDYVLGGLFPLGEAEEAGLRSRTRPSSPVCTRFSSNGLLWALAMKMAVEEINNKSDLLPGLRLGYDLFDTCSEPVVAMKPSLMFLAKAGSRDIAAYCNYTQYQPRVLAVIGPHSSELAMVTGKFFSFFLMPQVSYGASMELLSARETFPSFFRTVPSDRVQLTAAAELLQEFGWNWVAALGSDDEYGRQGLSIFSALAAARGICIAHEGLVPLPRADDSRLGKVQDVLHQVNQSSVQVVLLFASVHAAHALFNYSISSRLSPKVWVASEAWLTSDLVMGLPGMAQMGTVLGFLQRGAQLHEFPQYVKTHLALATDPAFCSALGEREQGLEEDVVGQRCPQCDCITLQNVSAGLNHHQTFSVYAAVYSVAQALHNTLQCNASGCPAQDPVKPWQLLENMYNLTFHVGGLPLRFDSSGNVDMEYDLKLWVWQGSVPRLHDVGRFNGSLRTERLKIRWHTSDNQKPVSRCSRQCQEGQVRRVKGFHSCCYDCVDCEAGSYRQNPDDIACTFCGQDEWSPERSTRCFRRRSRFLAWGEPAVLLLLLLLSLALGLVLAALGLFVHHRDSPLVQASGGPLACFGLVCLGLVCLSVLLFPGQPSPARCLAQQPLSHLPLTGCLSTLFLQAAEIFVESELPLSWADRLSGCLRGPWAWLVVLLAMLVEVALCTWYLVAFPPEVVTDWHMLPTEALVHCRTRSWVSFGLAHATNATLAFLCFLGTFLVRSQPGRYNRARGLTFAMLAYFITWVSFVPLLANVQVVLRPAVQMGALLLCVLGILAAFHLPRCYLLMRQPGLNTPEFFLGGGPGDAQGQNDGNTGNQGKHE

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 \(c\).

\(|P_{f(6CGZ_1)}(2) \setminus P_{f(7KLW_1)}(2)|=111\), \(|P_{f(7KLW_1)}(2) \setminus P_{f(6CGZ_1)}(2)|=59\). 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 1:100110101010110010101000111110011011010110011011100111001010111000001001110101101000111101000001000100101010010111100101001101011001011100001011100010000011011101011100010100100000011110110110110101111111010110011111100110010001111011111000111100100100110000001111000001001000001000
Pair \(Z_2\) Length of longest common subsequence
6CGZ_1,7KLW_1 170 4
6CGZ_1,9NOT_1 191 4
7KLW_1,9NOT_1 203 4

Newick tree

 
[
	9NOT_1:10.66,
	[
		6CGZ_1:85,7KLW_1:85
	]:17.66
]

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 \((0.85)(0.8)(\frac{476 }{\log_{20} 476}-\frac{194}{\log_{20}194})=82.2\)
Status Protein1 Protein2 d d1/2
Query variables 6CGZ_1 7KLW_1 109 91
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]