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Parikh vectors
4JLM_1 5LGB_1 4LUC_1 Letter Amino acid
13 21 9 Q Glutamine
13 43 11 G Glycine
12 14 4 H Histidine
28 58 12 L Leucine
12 28 6 F Phenylalanine
11 28 4 P Proline
12 7 8 Y Tyrosine
13 23 10 R Arginine
13 9 4 N Asparagine
11 22 15 D Aspartic acid
2 8 1 C Cysteine
29 41 13 E Glutamic acid
13 17 11 I Isoleucine
7 7 4 M Methionine
11 39 9 A Alanine
17 18 12 K Lycine
28 39 10 S Serine
15 30 12 T Threonine
7 10 0 W Tryptophan
13 35 15 V Valine 

4JLM_1|Chains A, B|Deoxycytidine kinase|Homo sapiens (9606)
>5LGB_1|Chain A|Peroxisomal N(1)-acetyl-spermine/spermidine oxidase,Peroxisomal N(1)-acetyl-spermine/spermidine oxidase|Mus musculus (10090)
>4LUC_1|Chains A, B|GTPase KRas|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 \(w=f(c)\)
4JLM , Knot 126 280 0.84 40 186 268 
MGSSHHHHHHSSGLVPRGSHMATPPKRSSPSFSASSEGTRIKKISIEGNIAAGKSTFVNILKQLSEDWEVVPEPVARWSNVQSTQDEFEELTMEQKNGGNVLQMMYEKPERWSFTFQTYACLSRIRAQLASLNGKLKDAEKPVLFFERSVYSDRYIFASNLYESESMNETEWTIYQDWHDWMNNQFGQSLELDGIIYLQATPETCLHRIYLRGRNEEQGIPLEYLEKLHYKHESWLLHRTLKTNFDYLQEVPILTLDVNEDFKDKYESLVEKVKEFLSTL
5LGB , Knot 199 497 0.82 40 240 463 
GGPGPRVLVVGSGIAGLGAAQKLCSHRAAPHLRVLEATASAGGRIRSERCFGGVVELGAHWIHGPSQDNPVFQLAAEFGLLGEKELSEENQLVDTGGHVALPSMIWSSSGTSVSLELMTEMARLFYGLIERTREFLNESETPMASVGEFLKKEISQQVASWTEDDEDTRKRKLAILNTFFNIECCVSGTHSMDLVALAPFGEYTVLPGLDCILAGGYQGLTDRILASLPKDTVAFDKPVKTIHWNGSFQEAAFPGETFPVLVECEDGARLPAHHVIVTVPLGFLKEHQDTFFEPPLPAKKAEAIKKLGFGTNNKIFLEFEEPFWEPDCQFIQVVWEDTSPLQDTALSLQDTWFKKLIGFLVQPSFESSHVLCGFIAGLESEFMETLSDEEVLLSLTQVLRRVTGNPQLPAAKSVRRSQWHSAPYTRGSYSYVAVGSTGDDLDLMAQPLPGLQVLFAGEATHRTFYSTTHGALLSGWREADRLVSLWDSQVEQSRPRL
4LUC , Knot 84 170 0.84 38 131 167 
GMTEYKLVVVGACGVGKSALTIQLIQNHFVDEYDPTIEDSYRKQVVIDGETSLLDILDTAGQEEYSAMRDQYMRTGEGFLLVFAINNTKSFEDIHHYREQIKRVKDSEDVPMVLVGNKSDLPSRTVDTKQAQDLARSYGIPFIETSAKTRQGVDDAFYTLVREIRKHKEK

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 \(c\).

\(|P_{f(4JLM_1)}(2) \setminus P_{f(5LGB_1)}(2)|=57\), \(|P_{f(5LGB_1)}(2) \setminus P_{f(4JLM_1)}(2)|=111\). 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 1:1100000000001111010011011000010101000100100101010111100011011001000101110111010010000001001010000110110110001001010100010100101011010101001001111100010000011100100000100001010001001100011001010111010101000100101010000011110010010000001110001000100100111101010001000000110010011001
Pair \(Z_2\) Length of longest common subsequence
4JLM_1,5LGB_1 168 4
4JLM_1,4LUC_1 181 3
5LGB_1,4LUC_1 179 3

Newick tree

 
[
	4LUC_1:91.91,
	[
		4JLM_1:84,5LGB_1:84
	]:7.91
]

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 \((0.85)(0.8)(\frac{777 }{\log_{20} 777}-\frac{280}{\log_{20}280})=136.\)
Status Protein1 Protein2 d d1/2
Query variables 4JLM_1 5LGB_1 175 137
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]