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Parikh vectors
9FAP_1 1WLA_1 5HOA_1 Letter Amino acid
4 0 6 C Cysteine
6 6 8 Q Glutamine
27 19 21 K Lycine
23 7 17 F Phenylalanine
35 7 31 V Valine
41 7 16 T Threonine
7 2 4 W Tryptophan
23 2 13 R Arginine
15 2 14 N Asparagine
20 8 17 D Aspartic acid
19 15 20 G Glycine
24 4 15 P Proline
24 5 19 S Serine
27 15 15 A Alanine
7 11 14 H Histidine
19 9 16 I Isoleucine
35 17 33 L Leucine
12 2 9 M Methionine
18 2 9 Y Tyrosine
19 13 15 E Glutamic acid 

9FAP_1|Chains A, D|Gamma-aminobutyric acid receptor subunit alpha-1|Homo sapiens (9606)
>1WLA_1|Chain A|MYOGLOBIN|Equus caballus (9796)
>5HOA_1|Chain A|Hepatocyte growth factor receptor|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 \(w=f(c)\)
9FAP , Knot 168 405 0.83 40 219 383 
TTVFTRILDRLLDGYDNRLRPGLGERVTEVKTDIFVTSFGPVSDHDMEYTIDVFFRQSWKDERLKFKGPMTVLRLNNLMASKIWTPDTFFHNGKKSVAHNMTMPNKLLRITEDGTLLYTMRLTVRAECPMHLEDFPMDAHACPLKFGSYAYTRAEVVYEWTREPARSVVVAEDGSRLNQYDLLGQTVDSGIVQSSTGEYVVMTTHFHLKRKIGYFVIQTYLPCIMTVILSQVSFWLNRESVPARTVFGVTTVLTMTTLSISARNSLPKVAYATAMDWFIAVCYAFVFSALIEFATVNYFTKRGYAWDGKSVVPEKPKKVKDPLIKKNNTYAPTATSYTPNLARGDPGLATIAKSATIEPKEVKPETKPPEPKKTFNSVSKIDRLSRIAFPLLFGIFNLVYWATYL
1WLA , Knot 75 153 0.82 38 115 148 
GLSDGEWQQVLNVWGKVEADIAGHGQEVLIRLFTGHPETLEKFDKFKHLKTEAEMKASEDLKKHGTVVLTALGGILKKKGHHEAELKPLAQSHATKHKIPIKYLEFISDAIIHVLHSKHPGDFGADAQGAMTKALELFRNDIAAKYKELGFQG
5HOA , Knot 139 312 0.85 40 203 302 
NTVHIDLSALNPELVQAVQHVVIGPSSLIVHFNEVIGRGHFGCVYHGTLLDNDGKKIHCAVKSLNRITDIGEVSQFLTEGIIMKDFSHPNVLSLLGICLRSEGSPLVVLPYMKHGDLRNFIRNETHNPTVKDLIGFGLQVAKGMKFVASKKFVHRDLAARNCMLDEKFTVKVADFGLARDMYDKEFDSVHNKTGAKLPVKWMALESLQTQKFTTKSDVWSFGVLLWELMTRGAPPYPDVNTFDITVYLLQGRRLLQPEYCPDPLYEVMLKCWHPKAEMRPSFSELVSRISAIFSTFIGEHYVHVNATYVNVK

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 \(c\).

\(|P_{f(9FAP_1)}(2) \setminus P_{f(1WLA_1)}(2)|=148\), \(|P_{f(1WLA_1)}(2) \setminus P_{f(9FAP_1)}(2)|=44\). 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 1:001100110011010000101111001001000111001111000010001011100010000101011101101001110011010011001000110010110011010001011001010101001101001110101011011001000101100100011001111001001000011100100111000010011100010100011011100011011011100101110000111001111001101001010100011011010110111110011110111011010010001011010011100100100111000000110100001011010111101100101010010100011010001001001001001111111111011011001
Pair \(Z_2\) Length of longest common subsequence
9FAP_1,1WLA_1 192 3
9FAP_1,5HOA_1 160 4
1WLA_1,5HOA_1 160 3

Newick tree

 
[
	1WLA_1:90.97,
	[
		9FAP_1:80,5HOA_1:80
	]:10.97
]

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 \((0.85)(0.8)(\frac{558 }{\log_{20} 558}-\frac{153}{\log_{20}153})=117.\)
Status Protein1 Protein2 d d1/2
Query variables 9FAP_1 1WLA_1 152 103.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]