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Parikh vectors
5DWA_1 3IFB_1 1FGU_1 Letter Amino acid
27 10 15 G Glycine
8 1 2 H Histidine
7 2 4 M Methionine
11 8 12 F Phenylalanine
11 10 17 T Threonine
2 2 4 W Tryptophan
19 6 16 A Alanine
14 13 15 E Glutamic acid
2 0 3 C Cysteine
12 7 17 I Isoleucine
24 10 17 L Leucine
15 0 8 P Proline
20 7 11 R Arginine
8 11 13 N Asparagine
9 4 7 Y Tyrosine
25 11 19 V Valine
9 2 7 Q Glutamine
23 5 25 S Serine
21 8 20 D Aspartic acid
11 14 20 K Lycine 

5DWA_1|Chains A, B|Type-2 restriction enzyme AgeI|Thalassobius gelatinovorus (53501)
>3IFB_1|Chain A|INTESTINAL FATTY ACID BINDING PROTEIN|Homo sapiens (9606)
>1FGU_1|Chains A, B|REPLICATION PROTEIN A 70 KDA DNA-BINDING SUBUNIT|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 \(w=f(c)\)
5DWA , Knot 121 278 0.81 40 176 270 
MRLDLDFGRGLVAHVMLDNVSEEQYQQISDYFVPLVNKPKLKSRDAIGQAFVMATEVCPDANPSDLWHHVLYRIYIREKIGTDPSQSWVRTSGEAFEVALVERYNPVLARHGIRLTALFKGQKGLALTRMGVADRVGSRKVDVMIEKQGGGRSPDAEGFGVVGGIHAKVSLAERVSDDIPASRIMMGEGLLSVLSTLDVKSFPPPHGDLVNRGELGTPDRPSDKRNYIEGHGDFSACFSYNLRTSPSNATTPSGRHIYVSGFSGQDDEFTDYLVAQLA
3IFB , Knot 65 131 0.80 36 102 127 
AFDSTWKVDRSENYDKFMEKMGVNIVKRKLAAHDNLKLTITQEGNKFTVKESSAFRNIEVVFELGVTFNYNLADGTELRGTWSLEGNKLIGKFKRTDNGNELNTVREIIGDELVQTYVYEGVEAKRIFKKD
1FGU , Knot 117 252 0.85 40 174 243 
MSKVVPIASLTPYQSKWTICARVTNKSQIRTWSNSRGEGKLFSLELVDESGEIRATAFNEQVDKFFPLIEVNKVYYFSKGTLKIANKQFTAVKNDYEMTFNNETSVMPCEDDHHLPTVQFDFTGIDDLENKSKDSLVDIIGICKSYEDATKITVRSNNREVAKRNIYLMDTSGKVVTATLWGEDADKFDGSRQPVLAIKGARVSDFGGRSLSVLSSSTIIANPDIPEAYKLRGWFDAEGQALDGVSISDLKS

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 \(c\).

\(|P_{f(5DWA_1)}(2) \setminus P_{f(3IFB_1)}(2)|=121\), \(|P_{f(3IFB_1)}(2) \setminus P_{f(5DWA_1)}(2)|=47\). 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 1:10101011011110111001000000010001111100101000011101111100101010100110011001010001100100011000101101111000011110011010111010011110011110011000101110001110010101111111101010110010001110011110111011001010011110101100101101001000000101010101010001000100100101001010110100001000111011
Pair \(Z_2\) Length of longest common subsequence
5DWA_1,3IFB_1 168 3
5DWA_1,1FGU_1 152 5
3IFB_1,1FGU_1 154 3

Newick tree

 
[
	3IFB_1:82.04,
	[
		5DWA_1:76,1FGU_1:76
	]: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 \((0.85)(0.8)(\frac{409 }{\log_{20} 409}-\frac{131}{\log_{20}131})=83.8\)
Status Protein1 Protein2 d d1/2
Query variables 5DWA_1 3IFB_1 108 78
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]