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Parikh vectors
5JYA_1 2HSZ_1 3EVU_1 Letter Amino acid
28 10 33 T Threonine
38 12 25 V Valine
14 3 19 R Arginine
19 13 19 N Asparagine
12 13 16 Q Glutamine
19 16 32 K Lycine
21 15 41 D Aspartic acid
21 11 38 E Glutamic acid
13 10 16 H Histidine
22 26 34 L Leucine
8 12 14 Y Tyrosine
33 18 22 A Alanine
36 18 45 G Glycine
8 4 20 M Methionine
11 15 20 F Phenylalanine
2 3 2 W Tryptophan
1 4 2 C Cysteine
19 16 22 I Isoleucine
11 12 12 P Proline
20 12 19 S Serine 

5JYA_1|Chains A, B, C, D|Glyceraldehyde-3-phosphate dehydrogenase|Streptococcus agalactiae (1311)
>2HSZ_1|Chains A, B|novel predicted phosphatase|Haemophilus somnus 129PT (205914)
>3EVU_1|Chain A|Myosin light chain kinase, Green fluorescent protein, Calmodulin-1  chimera|Aequorea victoria (6100)
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 \(w=f(c)\)
5JYA , Knot 152 356 0.83 40 203 330 
MGSSHHHHHHSSGLVPRGSHMVVKVGINGFGRIGRLAFRRIQNVEGVEVTRINDLTDPNMLAHLLKYDTTQGRFDGTVEVKEGGFEVNGQFVKVSAEREPANIDWATDGVEIVLEATGFFASKEKAEQHIHENGAKKVVITAPGGNDVKTVVFNTNHDILDGTETVISGASSTTNCLAPMAKALQDNFGVKQGLMTTIHAYTGDQMILDGPHRGGDLRRARAGAANIVPNSTGAAKAIGLVIPELNGKLDGAAQRVPVPTGSVTELVATLEKDVTVEEVNAAMKAAANDSYGYTEDPIVSSDIVGISYGSLFDATQTKVQTVDGNQLVKVVSWYDNEMSYTSQLVRTLEYFAKIAK
2HSZ , Knot 109 243 0.82 40 166 231 
MGSDKIHHHHHHENLYFQGMTQFKLIGFDLDGTLVNSLPDLALSINSALKDVNLPQASENLVMTWIGNGADVLSQRAVDWACKQAEKELTEDEFKYFKRQFGFYYGENLCNISRLYPNVKETLEALKAQGYILAVVTNKPTKHVQPILTAFGIDHLFSEMLGGQSLPEIKPHPAPFYYLCGKFGLYPKQILFVGDSQNDIFAAHSAGCAVVGLTYGYNYNIPIAQSKPDWIFDDFADILKITQ
3EVU , Knot 185 451 0.83 40 229 410 
MRGSHHHHHHGMASMTGGQQMGRDLYDDDDKDLATMVDSSRRKWNKTGHAVRAIGRLSSLENVYIMADKQKNGIKANFKIRHNIEDGGVQLAYHYQQNTPIGDGPVLLPDNHYLSTQSKLSKDPNEKRDHMVLLEFVTAAGITLGMDELYKGGTGGSMVSKGEELFTGVVPILVELDGDVNGHKFSVSGEGEGDATYGKLTLKFICTTGKLPVPWPTLVTTLTYGVQCFSRYPDHMKQHDFFKSAMPEGYIQERTIFFKDDGNYKTRAEVKFEGDTLVNRIELKGIDFKEDGNILGHKLEYNTRDQLTEEQIAEFKEAFSLFDKDGDGTITTKELGTVMRSLGQNPTEAELQDMINEVDADGNGTIDFPEFLTMMARKMKDTDSEEEIREAFRVFDKDGNGYISAAELRHVMTNLGEKLTDEEVDEMIREADIDGDGQVNYEEFVQMMTAK

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 \(c\).

\(|P_{f(5JYA_1)}(2) \setminus P_{f(2HSZ_1)}(2)|=95\), \(|P_{f(2HSZ_1)}(2) \setminus P_{f(5JYA_1)}(2)|=58\). 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 1:11000000000011110100111011101110110111001001011010010010010111011000000101010101001110101011010100011010110011011101011110000100010001100111011110010011100000110100011011000000111110110001110011100101001001110110011010010111101110001110111111101010101110011110101001110100010100101110111000010000111000111100101101000010010100110110100001000001100100110110
Pair \(Z_2\) Length of longest common subsequence
5JYA_1,2HSZ_1 153 6
5JYA_1,3EVU_1 140 7
2HSZ_1,3EVU_1 169 6

Newick tree

 
[
	2HSZ_1:83.83,
	[
		5JYA_1:70,3EVU_1:70
	]:13.83
]

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 \((0.85)(0.8)(\frac{599 }{\log_{20} 599}-\frac{243}{\log_{20}243})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 5JYA_1 2HSZ_1 124 104
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]