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Parikh vectors
5SCK_1 3VFO_1 4XQJ_1 Letter Amino acid
7 4 2 C Cysteine
35 19 6 L Leucine
7 14 5 Y Tyrosine
15 17 3 Q Glutamine
30 8 8 I Isoleucine
29 13 10 S Serine
19 22 2 T Threonine
59 24 2 A Alanine
20 18 6 D Aspartic acid
36 19 2 G Glycine
10 9 5 H Histidine
15 7 8 K Lycine
12 3 1 M Methionine
16 7 7 F Phenylalanine
36 26 8 R Arginine
8 6 8 N Asparagine
32 23 8 E Glutamic acid
21 16 0 P Proline
2 9 0 W Tryptophan
38 12 8 V Valine

5SCK_1|Chains A, B, C, D, E, F, G, H|Pyruvate kinase|Homo sapiens (9606)
>3VFO_1|Chain A|MHC class I antigen|Homo sapiens (9606)
>4XQJ_1|Chains A, D|Accessory gene regulator A|Staphylococcus aureus (strain COL) (93062)
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 \(w=f(c)\)
5SCK , Knot 179 447 0.81 40 214 412
GSMEGPAGYLRRADVAQLTQELGTAFFQQQQLPAAMADTFLEHLCLLDIDSEPVAARSTSIIATIGPASRSVERLKEMIKAGMNIARLNFSHGSHEYHAESIANVREAVESFAGSPLSYRPVAIALDTKGPGSGPGLSEQDVRDLRFGVEHGVDIVFASFVRKASDVAAVRAALGPEGHGIKIISKIENHEGVKRFDEILEVSDGIMVARGDLGIEIPAEKVFLAQKMMIGRCNLAGKPVVCATQMLESMITKPRPTRAETSDVANAVLDGADCIMLSGETAKGNFPVEAVKMQHAIAREAEAAVYHRQLFEELRRAAPLSRDPTEVTAIGAVEAAFKCCAAAIIVLTTTGRSAQLLSRYRPRAAVIAVTRSAQAARQVHLCRGVFPLLYREPPEAIWADDVDRRVQFGIESGKLRGFLRVGDLVIVVTGWRPGSGYTNIMRVLSIS
3VFO , Knot 120 276 0.81 40 182 264
GSHSMRYFYTAMSRPGRGEPRFIAVGYVDDTQFVRFDSDAASPRTEPRAPWIEQEGPEYWDRNTQIFKTNTQTYRESLRNLRGYYNQSEAGSHIIQRMYGCDLGPDGRLLRGHDQSAYDGKDYIALNEDLSSWTAADTAAQITQRKWEAARVAEQRAAYLEGLCVEWLRRYLENGKETLQRADPPKTHVTHHPVSDHEATLRCWALGFYPAEITLTWQRDGEDQTQDTELVETRPAGDRTFQKWAAVVVPSGEEQRYTCHVQHEGLPKPLTLRWEP
4XQJ , Knot 53 99 0.82 36 83 97
VETIELKRGSNSVYVQYDDIMFFESSTKSHRLIAHLDNRQIEFYGNLKELSQLDDRFFRCHNSFVVNRHNIESIDSKERIVYFKNKEHCYASVRNVKKI

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 \(c\).

\(|P_{f(5SCK_1)}(2) \setminus P_{f(3VFO_1)}(2)|=100\), \(|P_{f(3VFO_1)}(2) \setminus P_{f(5SCK_1)}(2)|=68\). 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 1:101011110100101101000110111000011111100110010110100011110000111011110001001001101110110101001000001001101001100111011000111111000111011110000100101110011011110110010011110111110101101100100001100100110100111110101110111001111001111000111011101001100110010100100001101110110011101001010111011010011100101110000110010011110001001011111011100011111110001001011000010111111000101100101001111110001101111001000101110010101110110111110110110100011011010
Pair \(Z_2\) Length of longest common subsequence
5SCK_1,3VFO_1 168 3
5SCK_1,4XQJ_1 193 3
3VFO_1,4XQJ_1 199 3

Newick tree

 
[
	4XQJ_1:10.25,
	[
		5SCK_1:84,3VFO_1:84
	]:18.25
]

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 \((0.85)(0.8)(\frac{723 }{\log_{20} 723}-\frac{276}{\log_{20}276})=123.\)
Status Protein1 Protein2 d d1/2
Query variables 5SCK_1 3VFO_1 157 126
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]