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Parikh vectors
5KJO_1 5URC_1 6LXR_1 Letter Amino acid
7 18 10 L Leucine
4 11 18 K Lycine
12 7 10 P Proline
9 4 8 N Asparagine
4 10 4 H Histidine
3 1 1 W Tryptophan
1 3 3 R Arginine
11 1 3 Q Glutamine
4 4 11 E Glutamic acid
2 2 5 M Methionine
11 7 14 F Phenylalanine
16 11 13 S Serine
4 3 3 Y Tyrosine
6 13 10 V Valine
6 8 7 D Aspartic acid
10 1 3 C Cysteine
30 7 26 G Glycine
3 0 17 I Isoleucine
18 9 12 T Threonine
19 21 10 A Alanine 

5KJO_1|Chain A|Endoglucanase V-like protein|Phanerochaete chrysosporium (2822231)
>5URC_1|Chains A, C|Hemoglobin subunit alpha|Homo sapiens (9606)
>6LXR_1|Chain A|Peptidyl-prolyl cis-trans isomerase|Trichomonas vaginalis (5722)
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 \(w=f(c)\)
5KJO , Knot 81 180 0.78 40 123 169 
ATGGYVQQATGQASFTMYSGCGSPACGKAASGFTAAINQLAFGSAPGLGAGDACGRCFALTGNHDPYSPNYTGPFGQTIVVKVTDLCPVQGNQEFCGQTTSNPTNQHGMPFHFDICEDTGGSAKFFPSGHGALTGTFTEVSCSQWSGSDGGQLWNGACLSGETAPNWPSTACGNKGTAPS
5URC , Knot 69 141 0.80 38 99 137 
VLSPADKTNVKAAWGKVGAHAGEYGAEALERMFLSFPTTKTYFPHFDLSHGSAQVKGHGKKVADALTNAVAHVDDMPNALSALSDLHAHKLRVDPVNFKLLSHCLLVTLAAHLPAEFTPAVHASLDKFLASVSTVLTSKYR
6LXR , Knot 88 188 0.81 40 130 183 
MLAFFATRVISAPKVTKKVFFKISINGEDAGTIKFGLFGDDVPKTAENFRALCTGEKGMGKLGKPLHYKGSPFHRVIPNFMIQGGDITSGNGYGGESIYGSKFADESFKITHDGPGLLSMANSGPNTNGSQFFITTVPCPWLNGKHVVFGKVIEGMEIVKKIESLGSQSGTPKAKIIIADCGEITELE

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 \(c\).

\(|P_{f(5KJO_1)}(2) \setminus P_{f(5URC_1)}(2)|=81\), \(|P_{f(5URC_1)}(2) \setminus P_{f(5KJO_1)}(2)|=57\). 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 1:101101001010101010010101101011011011100111101111111010100111010001001000111100111010010110100010100000100001111010100001101011101011101010010000101001101101101010011011001010010110
Pair \(Z_2\) Length of longest common subsequence
5KJO_1,5URC_1 138 3
5KJO_1,6LXR_1 137 3
5URC_1,6LXR_1 149 3

Newick tree

 
[
	5URC_1:72.87,
	[
		5KJO_1:68.5,6LXR_1:68.5
	]:4.37
]

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 \((0.85)(0.8)(\frac{321 }{\log_{20} 321}-\frac{141}{\log_{20}141})=55.2\)
Status Protein1 Protein2 d d1/2
Query variables 5KJO_1 5URC_1 68 60.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]