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Parikh vectors
5LBM_1 8FEK_1 5OKZ_1 Letter Amino acid
2 1 4 M Methionine
1 12 9 F Phenylalanine
2 33 14 P Proline
0 7 1 W Tryptophan
11 31 18 R Arginine
4 7 10 Q Glutamine
8 54 17 L Leucine
4 1 17 K Lycine
2 3 5 C Cysteine
2 6 8 Y Tyrosine
10 19 22 E Glutamic acid
4 62 30 G Glycine
7 28 22 V Valine
6 7 16 I Isoleucine
6 19 18 S Serine
4 29 25 T Threonine
9 61 23 A Alanine
2 4 16 N Asparagine
6 35 18 D Aspartic acid
1 15 2 H Histidine 

5LBM_1|Chains A, B, C, D|Transcriptional repressor FrmR|Escherichia coli O157:H7 (83334)
>8FEK_1|Chain A|PBP cyclase Ulm16|Streptomyces sp. KCB13F003 (2052824)
>5OKZ_1|Chains A[auth I], DA[auth S], J[auth c], T[auth m]|Exosome complex component CSL4|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
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 \(w=f(c)\)
5LBM , Knot 46 91 0.76 38 75 89 
MPSTPEEKKKVLTRVRRIRGQIDALERSLEGDAECRAILQQIAAVRGAANGLMAEVLESHIRETFDRNDCYSREVSQSVDDTIELVRAYLK
8FEK , Knot 164 434 0.76 40 173 380 
GPLGSGAGDGAPVGLDLDRLARDCDVVGGQLALHHQGTLTTWEFGTEEHAGGRPVHVGSAFPYGSVTKAFTATAVLQLAGDGDLDLDRPVRELLPEAEAEAEIEVEAGSGTGARADGGHPALAATLRQLLSHTAGLPSDHDDERAPSLRRWLTGFLALPVGPWPAPGSFSYSNVGYGIAGRVVEAVTGLTWSEAVRDFLLHPLGTAITVLPTDPGSLPAGGLAGSAADLVRLGRLHLDEPGDPDLARLADPDALREMARPTAGADPFGLADGWGPGLGRFGPAGNRWLGHDGTLDGATCHLRIHPGRGTVVALTTNSPTGQALWDAVVDALRDADIDVGVHRPAPPPAIAAAAFADCTGTYRNGDLAVTVGIDGPYLVLELPGGARELAQPLAHRTFSSRGAGFLGRFVTDARSDAVHALQYSGRTLLREAGES
5OKZ , Knot 132 295 0.84 40 181 276 
GPHMACNFQFPEIAYPGKLICPQYGTENKDGEDIIFNYVPGPGTKLIQYEHNGRTLEAITATLVGTVRCEEEKKTDQEEEREGTDQSTEEEKSVDASPNDVTRRTVKNILVSVLPGTEKGRKTNKYANNDFANNLPKEGDIVLTRVTRLSLQRANVEILAVEDKPSPIDSGIGSNGSGIVAAGGGSGAATFSVSQASSDLGETFRGIIRSQDVRSTDRDRVKVIECFKPGDIVRAQVLSLGDGTNYYLTTARNDLGVVFARAANGAGGLMYATDWQMMTSPVTGATEKRKCAKPF

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 \(c\).

\(|P_{f(5LBM_1)}(2) \setminus P_{f(8FEK_1)}(2)|=28\), \(|P_{f(8FEK_1)}(2) \setminus P_{f(5LBM_1)}(2)|=126\). 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 1:1100100000110010010101011000101010001110011110111011110110001000100000000010001000101101010
Pair \(Z_2\) Length of longest common subsequence
5LBM_1,8FEK_1 154 3
5LBM_1,5OKZ_1 152 5
8FEK_1,5OKZ_1 170 4

Newick tree

 
[
	8FEK_1:82.72,
	[
		5LBM_1:76,5OKZ_1:76
	]:6.72
]

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 \((0.85)(0.8)(\frac{525 }{\log_{20} 525}-\frac{91}{\log_{20}91})=129.\)
Status Protein1 Protein2 d d1/2
Query variables 5LBM_1 8FEK_1 154 93.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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