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Parikh vectors
5SGM_1 3ZFD_1 6XYU_1 Letter Amino acid
16 24 52 G Glycine
18 29 21 K Lycine
12 4 18 M Methionine
7 0 15 W Tryptophan
14 11 34 P Proline
16 25 34 V Valine
15 17 33 D Aspartic acid
12 7 8 C Cysteine
13 11 14 H Histidine
21 23 28 I Isoleucine
36 31 39 L Leucine
22 21 32 T Threonine
12 9 25 Y Tyrosine
14 19 27 R Arginine
14 20 33 N Asparagine
24 26 34 E Glutamic acid
15 12 26 F Phenylalanine
24 30 36 S Serine
21 20 51 A Alanine
17 12 21 Q Glutamine 

5SGM_1|Chain A|cAMP and cAMP-inhibited cGMP 3',5'-cyclic phosphodiesterase 10A|Homo sapiens (9606)
>3ZFD_1|Chain A|CHROMOSOME-ASSOCIATED KINESIN KIF4|MUS MUSCULUS (10090)
>6XYU_1|Chain A|Acetylcholinesterase|Drosophila melanogaster (7227)
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 \(w=f(c)\)
5SGM , Knot 152 343 0.86 40 221 334 
GSSICTSEEWQGLMQFTLPVRLCKEIELFHFDIGPFENMWPGIFVYMVHRSCGTSCFELEKLCRFIMSVKKNYRRVPYHNWKHAVTVAHCMYAILQNNHTLFTDLERKGLLIACLCHDLDHRGFSNSYLQKFDHPLAALYSTSTMEQHHFSQTVSILQLEGHNIFSTLSSSEYEQVLEIIRKAIIATDLALYFGNRKQLEEMYQTGSLNLNNQSHRDRVIGLMMTACDLCSVTKLWPVTKLTANDIYAEFWAEGDEMKKLGIQPIPMMDRDKKDEVPQGQLGFYNAVAIPCYTTLTQILPPTEPLLKACRDNLSQWEKVIRGEETATWISSPSVAQKAAASED
3ZFD , Knot 149 351 0.83 38 203 332 
MKEEVKGIPVRVALRCRPLVSKEIKEGCQTCLSFVPGEPQVVVGNDKSFTYDFVFDPSTEQEEVFNTAVAPLIKGVFKGYNATVLAYGQTGSGKTYSMGGAYTAEQEHDSAIGVIPRVIQLLFKEINKKSDFEFTLKVSYLEIYNEEILDLLCSSREKATQINIREDPKEGIKIVGLTEKTVLVASDTVSCLEQGNNSRTVASTAMNSQSSRSHAIFTISIEQRKKNDKNSSFRSKLHLVDLAGSERQKKTKAEGDRLREGININRGLLCLGNVISALGDDKKGNFVPYRDSKLTRLLQDSLGGNSHTLMIACVSPADSNLEETLNTLRYADRARKIKNKPIINHHHHHHH
6XYU , Knot 236 581 0.86 40 281 549 
VIDRLVVQTSSGPVRGRSVTVQGREVHVYTGIPYAKPPVEDLRFRKPVPAEPWHGVLDATGLSATCVQERYEYFPGFSGEEIWNPNTNVSEDCLYINVWAPAKARLRHGRGANGGEHPNGKQADTDHLIHNGNPQNTTNGLPILIWIYGGGFMTGSATLDIYNADIMAAVGNVIVASFQYRVGAFGFLHLAPEMPSEFAEEAPGNVGLWDQALAIRWLKDNAHAFGGNPEWMTLFGESAGSSSVNAQLMSPVTRGLVKRGMMQSGTMNAPWSHMTSEKAVEIGKALINDCNCNASMLKTNPAHVMSCMRSVDAKTISVQQWNSYSGILSFPSAPTIDGAFLPADPMTLMKTADLKDYDILMGNVRDEGTYFLLYDFIDYFDKDDATALPRDKYLEIMNNIFGKATQAEREAIIFQYTSWEGNPGYQNQQQIGRAVGDHFFTCPTNEYAQALAERGASVHYYYFTHRTSTSLWGEWMGVLHGDEIEYFFGQPLNNSLQYRPVERELGKRMLSAVIEFAKTGNPAQDGEEWPNFSKEDPVYYIFSTDDKIEKLARGPLAARCSFWNDYLPKVRSWAGTCDGDS

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 \(c\).

\(|P_{f(5SGM_1)}(2) \setminus P_{f(3ZFD_1)}(2)|=90\), \(|P_{f(3ZFD_1)}(2) \setminus P_{f(5SGM_1)}(2)|=72\). 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 1:1001000001011101011101000101101011110011111110110000100010100100111010000001100010011011001011100000110010001111101000100011000010010011111000001000010001011010100110010000000110110011110011101100001001000101010000000011111101001001001111001010010101110100100111011111000000011010111001111100001001111001110100001001001101000101100101100111000
Pair \(Z_2\) Length of longest common subsequence
5SGM_1,3ZFD_1 162 4
5SGM_1,6XYU_1 176 4
3ZFD_1,6XYU_1 160 4

Newick tree

 
[
	5SGM_1:86.04,
	[
		3ZFD_1:80,6XYU_1:80
	]:6.04
]

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 \((0.85)(0.8)(\frac{694 }{\log_{20} 694}-\frac{343}{\log_{20}343})=96.3\)
Status Protein1 Protein2 d d1/2
Query variables 5SGM_1 3ZFD_1 122 121
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]