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Parikh vectors
4KTS_1 5HNI_1 7UCU_1 Letter Amino acid
14 21 30 K Lycine
3 17 18 F Phenylalanine
8 15 20 P Proline
10 9 4 Y Tyrosine
16 14 8 N Asparagine
6 17 17 D Aspartic acid
3 14 7 H Histidine
4 15 21 E Glutamic acid
25 20 38 G Glycine
34 19 26 S Serine
14 15 28 A Alanine
2 13 12 R Arginine
10 8 8 Q Glutamine
4 4 2 W Tryptophan
12 6 14 C Cysteine
14 34 25 L Leucine
10 16 24 T Threonine
15 16 24 I Isoleucine
2 9 9 M Methionine
17 30 39 V Valine 

4KTS_1|Chain A|Cationic trypsin|Bos taurus (9913)
>5HNI_1|Chains A[auth X], B[auth Y]|Hepatocyte growth factor receptor|Homo sapiens (9606)
>7UCU_1|Chains A, B|Alcohol dehydrogenase E chain|Equus caballus (9796)
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 \(w=f(c)\)
4KTS , Knot 101 223 0.81 40 140 216 
IVGGYTCGANTVPYQVSLNSGYHFCGGSLINSQWVVSAAHCYKSGIQVRLGEDNINVVEGNEQFISASKSIVHPSYNSNTLNNDIMLIKLKSAASLNSRVASISLPTSCASAGTQCLISGWGNTKSSGTSYPDVLKCLKAPILSDSSCKSAYPGQITSNMFCAGYLEGGKDSCQGDSGGPVVCSGKLQGIVSWGSGCAQKNKPGVYTKVCNYVSWIKQTIASN
5HNI , Knot 140 312 0.86 40 203 303 
NTVHIDLSALNPELVQAVQHVVIGPSSLIVHFNEVIGRGHFGCVYHGTLLDNDGKKIHCAVKSLNRITDIGEVSQFLTEGIIMKDFSHPNVLSLLGICLRSEGSPLVVLPYMKHGDLRNFIRNETHNPTVKDLIGFGLQVAKGMKFLASKKFVHRDLAARNCMLDEKFTVKVADFGLARDMYDKEFDSVHNKTGAKLPVKWMALESLQTQKFTTKSDVWSFGVLLWELMTRGAPPYPDVNTFDITVYLLQGRRLLQPEYCPDPLYEVMLKCWHPKAEMRPSFSELVSRISAIFSTFIGEHYVHVNATYVNVK
7UCU , Knot 158 374 0.83 40 210 358 
STAGKVIKCKAAVLWEEKKPFSIEEVEVAPPKAHEVRIKMVATGICRSDDHVVSGTLVTPLPVIAGHEAAGIVESIGEGVTTVRPGDKVIPLFTPQCGKCRVCKHPEGNFCLKNDLSMPRGTMQDGTSRFTCRGKPIHHFLGTSTFSQYTVVDEISVAKIDAASPLEKVCLIGCGFSTGYGSAVKVAKVTQGSTCAVFGLGGVGLSVIMGCKAAGAARIIGVDINKDKFAKAKEVGATECVNPQDYKKPIQEVLTEMSNGGVDFSFEVIGRLDTMVTALSCCQEAYGVSVIVGVPPDSQNLSMNPMLLLSGRTWKGAIFGGFKSKDSVPKLVADFMAKKFALDPLITHVLPFEKINEGFDLLRSGESIRTILTF

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 \(c\).

\(|P_{f(4KTS_1)}(2) \setminus P_{f(5HNI_1)}(2)|=60\), \(|P_{f(5HNI_1)}(2) \setminus P_{f(4KTS_1)}(2)|=123\). 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 1:1111000110011001010010010110110001110110000011010110001011010001101000110100000010001111010011010001101011000101100011011100000100010110010111100000001011010001101101011000001001111100101011101101010000111000100010110001100
Pair \(Z_2\) Length of longest common subsequence
4KTS_1,5HNI_1 183 4
4KTS_1,7UCU_1 170 3
5HNI_1,7UCU_1 161 5

Newick tree

 
[
	4KTS_1:90.76,
	[
		7UCU_1:80.5,5HNI_1:80.5
	]:10.26
]

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 \((0.85)(0.8)(\frac{535 }{\log_{20} 535}-\frac{223}{\log_{20}223})=89.4\)
Status Protein1 Protein2 d d1/2
Query variables 4KTS_1 5HNI_1 120 100
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]