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Parikh vectors
1KXJ_1 9DSN_1 3EHT_1 Letter Amino acid
3 5 6 C Cysteine
22 21 35 E Glutamic acid
19 59 38 L Leucine
11 9 17 F Phenylalanine
13 36 22 S Serine
19 52 27 V Valine
8 30 26 D Aspartic acid
8 16 30 N Asparagine
2 15 14 Q Glutamine
7 22 12 H Histidine
13 24 25 I Isoleucine
1 4 10 W Tryptophan
18 45 10 R Arginine
10 3 39 K Lycine
5 7 7 M Methionine
4 39 22 T Threonine
22 55 35 G Glycine
8 50 25 P Proline
8 7 20 Y Tyrosine
4 75 56 A Alanine 

1KXJ_1|Chains A, B|Amidotransferase hisH|Thermotoga maritima (2336)
>9DSN_1|Chains A, B, C, D|2-succinyl-5-enolpyruvyl-6-hydroxy-3-cyclohexene-1-carboxylate synthase|Mycobacterium tuberculosis H37Rv (83332)
>3EHT_1|Chain A|FUSION PROTEIN OF CRFR1 EXTRACELLULAR DOMAIN AND MBP|Escherichia coli (83333)
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 \(w=f(c)\)
1KXJ , Knot 94 205 0.81 40 138 196 
GHMRIGIISVGPGNIMNLYRGVKRASENFEDVSIELVESPRNDLYDLLFIPGVGHFGEGMRRLRENDLIDFVRKHVEDERYVVGVCLGMQLLFEESEEAPGVKGLSLIEGNVVKLRSRRLPHMGWNEVIFKDTFPNGYYYFVHTYRAVCEEEHVLGTTEYDGEIFPSAVRKGRILGFQFHPEKSSKIGRKLLEKVIECSLSRRGS
9DSN , Knot 216 574 0.79 40 212 509 
MGSSHHHHHHSSGLVPRGSHMNPSTTQARVVVDELIRGGVRDVVLCPGSRNAPLAFALQDADRSGRIRLHVRIDERTAGYLAIGLAIGAGAPVCVAMTSGTAVANLGPAVVEANYARVPLIVLSANRPYELLGTGANQTMEQLGYFGTQVRASISLGLAEDAPERTSALNATWRSATCRVLAAATGARTANAGPVHFDIPLREPLVPDPEPLGAVTPPGRPAGKPWTYTPPVTFDQPLDIDLSVDTVVISGHGAGVHPNLAALPTVAEPTAPRSGDNPLHPLALPLLRPQQVIMLGRPTLHRPVSVLLADAEVPVFALTTGPRWPAVSGNSQATGTRAVTTGAPRPAWLDRCAAMNRHAIAAVREQLAAHPLTTGLHVAAAVSHALRPGDQLVLGASNPVRDVALAGLDTRGIRVRSNRGVAGIDGTVSTAIGAALAYEGAHERTGSPDSPPRTIALIGDLTFVHDSSGLLIGPTEPIPRSLTIVVSNDNGGGIFELLEQGDPRFSDVSSRIFGTPHDVDVGALCRAYHVESRQIEVDELGPTLDQPGAGMRVLEVKADRSSLRQLHAAIKAAL
3EHT , Knot 189 476 0.81 40 235 434 
MAKIEEGKLVIWINGDKGYNGLAEVGKKFEKDTGIKVTVEHPDKLEEKFPQVAATGDGPDIIFWAHDRFGGYAQSGLLAEITPDKAFQDKLYPETWDAVRYNGKLIAYPIAVEALSLIYNKDLLPNPPKTWEEIPALDKELKAKGKSALMFNLQEPYFTWPLIAADGGYAFKYENGKYDIKDVGVDNAGAKAGLTFLVDLIKNKHMNADTDYSIAEAAFNKGETAMTINGPWAWSNIDTSKVNYGVTVLPTFKGQPSKPFVGVLSAGINAASPNKELAKEFLENYLLTDEGLEAVNKDKPLGAVALKSYEEELAKDPRIAATMENAQKGEIMPNIPQMSAFWYAVRTAVINAASGRQTVDEALKDAQTNAAAEFSLQDQHCESLSLASNISGLQCNASVDLIGTCWPRSPAGQLVVRPCPAFFYGVRYNTTNNGYRECLANGSWAARVNYSECQEILNEEKKSKVHYHVAHHHHHH

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 \(c\).

\(|P_{f(1KXJ_1)}(2) \setminus P_{f(9DSN_1)}(2)|=48\), \(|P_{f(9DSN_1)}(2) \setminus P_{f(1KXJ_1)}(2)|=122\). 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 1:1010111101111011010011001000100101011001000100111111110110110010000110110001000001111011101110000011110110110101101000011011100111000110100011000011000001110000010111011001011110101000001100110011000100010
Pair \(Z_2\) Length of longest common subsequence
1KXJ_1,9DSN_1 170 3
1KXJ_1,3EHT_1 183 4
9DSN_1,3EHT_1 157 6

Newick tree

 
[
	1KXJ_1:91.34,
	[
		9DSN_1:78.5,3EHT_1:78.5
	]:12.84
]

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 \((0.85)(0.8)(\frac{779 }{\log_{20} 779}-\frac{205}{\log_{20}205})=159.\)
Status Protein1 Protein2 d d1/2
Query variables 1KXJ_1 9DSN_1 200 135.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]