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Parikh vectors
1SSH_1 6EUU_1 8SBE_1 Letter Amino acid
3 2 20 R Arginine
3 14 15 N Asparagine
4 25 42 L Leucine
0 4 17 M Methionine
6 17 36 S Serine
4 12 39 A Alanine
1 13 10 Q Glutamine
3 13 31 F Phenylalanine
3 5 24 P Proline
4 8 11 D Aspartic acid
3 20 46 I Isoleucine
4 23 20 K Lycine
2 14 31 T Threonine
2 3 11 W Tryptophan
5 11 39 V Valine
0 4 12 C Cysteine
4 15 32 E Glutamic acid
7 15 52 G Glycine
0 3 20 H Histidine
2 18 23 Y Tyrosine 

1SSH_1|Chain A|Hypothetical 40.4 kDa protein in PES4-HIS2 intergenic region|Saccharomyces cerevisiae (4932)
>6EUU_1|Chains A, B|Listeriolysin positive regulatory factor A|Listeria monocytogenes (1639)
>8SBE_1|Chain A|Vesicular glutamate transporter 2|Rattus norvegicus (10116)
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 \(w=f(c)\)
1SSH , Knot 35 60 0.79 34 57 58 
GSSPKAVALYSFAGEESGDLPFRKGDVITILKKSDSQNDWWTGRVNGREGIFPANYVELV
6EUU , Knot 104 239 0.79 40 157 229 
GAMNAQAEEFKKYLETNGIKPKQFHKKELIFNQWDPQEYCIFLYDGITKLTSISENGTIMNLQYYKGAFVIMSGFIDTETSVGYYNLEVISEQATAYVIKINELKELLSKNLTHFFYVFQTLQKQVSYSLAKFNDFSINGKLGSICGQLLILTYVYGKETPDGIKITLDNLTMQELGYSSGIAHSSAVSRIISKLKQEKVIVYKNSCFYVQNLDYLKRYAPKLDEWFYLACPATWGKLN
8SBE , Knot 207 531 0.81 40 253 490 
METIELTEDGKPLEVPEKKAPLCDCTCFGLPRRYIIAIMSGLGFCISFGIRCNLGVAIVDMVNNSTIHRGGKVIKEKAKFNWDPETVGMIHGSFFWGYIITQIPGGYIASRLAANRVFGAAILLTSTLNMLIPSAARVHYGCVIFVRILQGLVEGVTYPACHGIWSKWAPPLERSRLATTSFCGSYAGAVIAMPLAGILVQYTGWSSVFYVYGSFGMVWYMFWLLVSYESPAKHPTITDEERRYIEESIGESANLLGAMEKFKTPWRKFFTSMPVYAIIVANFCRSWTFYLLLISQPAYFEEVFGFEISKVGMLSAVPHLVMTIIVPIGGQIADFLRSKQILSTTTVRKIMNCGGFGMEATLLLVVGYSHTRGVAISFLVLAVGFSGFAISGFNVNHLDIAPRYASILMGISNGVGTLSGMVCPIIVGAMTKNKSREEWQYVFLIAALVHYGGVIFYALFASGEKQPWADPEETSEEKCGFIHEDELDEETGDITQNYINYGTTKSYGATGRPLEVLFQGPHHHHHHHHHH

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 \(c\).

\(|P_{f(1SSH_1)}(2) \setminus P_{f(6EUU_1)}(2)|=34\), \(|P_{f(6EUU_1)}(2) \setminus P_{f(1SSH_1)}(2)|=134\). 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 1:100101111001110001011100101101100000000110101010011111001011
Pair \(Z_2\) Length of longest common subsequence
1SSH_1,6EUU_1 168 2
1SSH_1,8SBE_1 224 3
6EUU_1,8SBE_1 200 4

Newick tree

 
[
	8SBE_1:11.59,
	[
		1SSH_1:84,6EUU_1:84
	]:28.59
]

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 \((0.85)(0.8)(\frac{299 }{\log_{20} 299}-\frac{60}{\log_{20}60})=76.9\)
Status Protein1 Protein2 d d1/2
Query variables 1SSH_1 6EUU_1 96 60.5
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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