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Parikh vectors
4BVS_1 1HEO_1 1WFI_1 Letter Amino acid
5 8 0 C Cysteine
24 8 12 L Leucine
8 3 2 F Phenylalanine
23 5 5 I Isoleucine
21 7 6 T Threonine
49 12 2 A Alanine
5 3 3 Q Glutamine
22 2 11 E Glutamic acid
36 10 15 S Serine
1 6 4 W Tryptophan
33 7 15 V Valine
37 12 10 G Glycine
22 1 2 H Histidine
7 6 10 K Lycine
14 2 2 M Methionine
17 2 7 P Proline
26 11 8 R Arginine
6 14 7 N Asparagine
21 7 8 D Aspartic acid
6 3 2 Y Tyrosine 

4BVS_1|Chains A, B|CYANURIC ACID AMIDOHYDROLASE|PSEUDOMONAS SP. ADP (47660)
>1HEO_1|Chain A|HEN EGG WHITE LYSOZYME|Gallus gallus (9031)
>1WFI_1|Chain A|nuclear distribution gene C homolog|Mus musculus (10090)
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 \(w=f(c)\)
4BVS , Knot 152 383 0.78 40 199 351 
MGSSHHHHHHSSGLVPRGSHMYHIDVFRIPCHSPGDTSGLEDLIETGRVAPADIVAVMGKTEGNGCVNDYTREYATAMLAACLGRHLQLPPHEVEKRVAFVMSGGTEGVLSPHHTVFARRPAIDAHRPAGKRLTLGIAFTRDFLPEEIGRHAQITETAGAVKRAMRDAGIASIDDLHFVQVKCPLLTPAKIASARSRGCAPVTTDTYESMGYSRGASALGIALATEEVPSSMLVDESVLNDWSLSSSLASASAGIELEHNVVIAIGMSEQATSELVIAHGVMSDAIDAASVRRTIESLGIRSDDEMDRIVNVFAKAEASPDGVVRGMRHTMLSDSDINSTRHARAVTGAAIASVVGHGMVYVSGGAEHQGPAGGGPFAVIARA
1HEO , Knot 67 129 0.84 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGVLQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
1WFI , Knot 65 131 0.80 38 98 122 
GSSGSSGPNYRWTQTLAELDLAVPFRVSFRLKGKDVVVDIQRRHLRVGLKGQPPVVDGELYNEVKVEESSWLIEDGKVVTVHLEKINKMEWWNRLVTSDPEINTKKINPENSKLSDLDSETRSMVSGPSSG

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 \(c\).

\(|P_{f(4BVS_1)}(2) \setminus P_{f(1HEO_1)}(2)|=136\), \(|P_{f(1HEO_1)}(2) \setminus P_{f(4BVS_1)}(2)|=41\). 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 1:11000000000011110100100101101100011000110011001011110111111000101010000000101111101100101110010001111101100111010001110011101001110010111110001110011001010001111001100111101001011010011101101101000101110000000110001101111111000110011100011001010001101011101000111111100010001111011100110110100010011100000100110111010101011101100011000010000010110111110111011101011100011111111111101
Pair \(Z_2\) Length of longest common subsequence
4BVS_1,1HEO_1 177 3
4BVS_1,1WFI_1 181 3
1HEO_1,1WFI_1 146 3

Newick tree

 
[
	4BVS_1:94.36,
	[
		1HEO_1:73,1WFI_1:73
	]:21.36
]

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 \((0.85)(0.8)(\frac{512 }{\log_{20} 512}-\frac{129}{\log_{20}129})=113.\)
Status Protein1 Protein2 d d1/2
Query variables 4BVS_1 1HEO_1 142 95
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]