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Parikh vectors
7TPM_1 5CCB_1 4NFD_1 Letter Amino acid
12 10 3 D Aspartic acid
0 6 1 C Cysteine
4 20 11 R Arginine
17 11 7 I Isoleucine
19 23 9 L Leucine
8 17 6 P Proline
0 5 4 W Tryptophan
11 5 6 Y Tyrosine
11 28 7 V Valine
3 5 2 M Methionine
9 21 2 A Alanine
15 2 4 N Asparagine
8 12 7 Q Glutamine
8 17 6 E Glutamic acid
15 30 9 G Glycine
6 13 4 H Histidine
18 6 5 K Lycine
14 11 7 F Phenylalanine
19 21 13 S Serine
11 26 7 T Threonine 

7TPM_1|Chain A|Outer membrane lipoprotein carrier protein LolA|Borreliella burgdorferi (139)
>5CCB_1|Chain A|tRNA (adenine(58)-N(1))-methyltransferase catalytic subunit TRMT61A|Homo sapiens (9606)
>4NFD_1|Chain A|Paired immunoglobulin-like type 2 receptor beta|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 \(w=f(c)\)
7TPM , Knot 94 208 0.80 36 141 194 
MQISANQYFEGIYAKYQNIEDMQATINFTLKGLKQTGVLLYKFPDKFIINLDSNNQVFVSDGEFLTVYVPSLGTSFNQQLLKGSSGGGLMKVLNSEYSVSYTNSPNLEDLDSSEPGKYIKLTFSRKLYKGAATINSFIIAFAPDGIIRRITAFPTSGGREIVIDLTAVKFNVGILDSKFKYDPPKSSNKVDNFLYDIKKNLEHHHHHH
5CCB , Knot 127 289 0.83 40 177 276 
MSFVAYEELIKEGDTAILSLGHGAMVAVRVQRGAQTQTRHGVLRHSVDLIGRPFGSKVTCGRGGWVYVLHPTPELWTLNLPHRTQILYSTDIALITMMLELRPGSVVCESGTGSGSVSHAIIRTIAPTGHLHTVEFHQQRAEKAREEFQEHRVGRWVTVRTQDVCRSGFGVSHVADAVFLDIPSPWEAVGHAWDALKVEGGRFCSFSPCIEQVQRTCQALAARGFSELSTLEVLPQVYNVRTVSLPPPDLGTGTDGPAGSDTSPFRSGTPMKEAVGHTGYLTFATKTPG
4NFD , Knot 64 120 0.85 40 103 116 
MLYGVTQPKHLSASMGGSVEIPFSFYYPWELAIVPNVRISWRRGHFHGQSFYSTRPPSIHKDYVNRLFLNWTEGQESGFLRISNLRKEDQSVYFCRVELDTRRSGRQQWQSIKGTKLTIT

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 \(c\).

\(|P_{f(7TPM_1)}(2) \setminus P_{f(5CCB_1)}(2)|=69\), \(|P_{f(5CCB_1)}(2) \setminus P_{f(7TPM_1)}(2)|=105\). 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 1:1010100010110100001001010101010110001111001100111010000011100101101011011001000110100111110110000010000010100100001100101010001001110100111111101110010111001100111010110101111000100011000001001100100010000000
Pair \(Z_2\) Length of longest common subsequence
7TPM_1,5CCB_1 174 3
7TPM_1,4NFD_1 162 3
5CCB_1,4NFD_1 162 3

Newick tree

 
[
	5CCB_1:85.04,
	[
		7TPM_1:81,4NFD_1:81
	]:4.04
]

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 \((0.85)(0.8)(\frac{497 }{\log_{20} 497}-\frac{208}{\log_{20}208})=83.6\)
Status Protein1 Protein2 d d1/2
Query variables 7TPM_1 5CCB_1 107 91.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]