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Parikh vectors
4FIZ_1 6SYD_1 1XMZ_1 Letter Amino acid
12 11 1 R Arginine
15 7 0 D Aspartic acid
1 8 2 C Cysteine
7 3 1 Q Glutamine
12 6 4 I Isoleucine
16 6 6 K Lycine
3 2 3 M Methionine
22 10 5 S Serine
33 6 2 V Valine
11 14 2 N Asparagine
27 12 10 G Glycine
16 8 3 L Leucine
12 3 6 F Phenylalanine
8 2 4 P Proline
14 7 7 T Threonine
2 6 0 W Tryptophan
27 12 2 A Alanine
14 2 6 E Glutamic acid
12 1 8 H Histidine
6 3 1 Y Tyrosine 

4FIZ_1|Chain A|17beta-hydroxysteroid dehydrogenase|Cochliobolus lunatus (5503)
>6SYD_1|Chain A|Lysozyme|Gallus gallus (9031)
>1XMZ_1|Chains A, C[auth B]|GFP-like non-fluorescent chromoprotein FP595 chain 1|Anemonia sulcata (6108)
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 \(w=f(c)\)
4FIZ , Knot 120 270 0.83 40 167 254 
MPHVENASETYIPGRLDGKVALVTGSGRGIGAAVAVHLGRLGAKVVVNYANSTKDAEKVVSEIKALGSDAIAIKADIRQVPEIVKLFDQAVAHFGHLDIAVSNSGVVSFGHLKDVTEEEFDRVFSLNTRGQFFVAREAYRHLTEGGRIVLTSSNTSKDFSVPKHSLYSGSKGAVDSFVRIFSKDCGDKKITVNAVAPGGTVTDMFHEVSHHYIPNGTSYTAEQRQQMAAHASPLHRNGWPQDVANVVGFLVSKEGEWVNGKVLTLDGGAA
6SYD , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
1XMZ , Knot 38 74 0.73 38 58 67 
MRGSHHHHHHGSASFLKKTMPFKTTIEGTVNGHYFKCTGKGEGNPFEGTQEMKIEVIEGGPLPFAFHILSTSCX

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 \(c\).

\(|P_{f(4FIZ_1)}(2) \setminus P_{f(6SYD_1)}(2)|=119\), \(|P_{f(6SYD_1)}(2) \setminus P_{f(4FIZ_1)}(2)|=56\). 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 1:110100100001110101011110101011111111011011101110010000010011001011100111101010011011011001110110101110001110110100100001001101000101111001000100110111000000001011000100100111001101100001000101011111101001100100001101000010000011101011000111001101111110001011010110101111
Pair \(Z_2\) Length of longest common subsequence
4FIZ_1,6SYD_1 175 3
4FIZ_1,1XMZ_1 169 3
6SYD_1,1XMZ_1 126 2

Newick tree

 
[
	4FIZ_1:92.41,
	[
		1XMZ_1:63,6SYD_1:63
	]:29.41
]

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 \((0.85)(0.8)(\frac{399 }{\log_{20} 399}-\frac{129}{\log_{20}129})=81.6\)
Status Protein1 Protein2 d d1/2
Query variables 4FIZ_1 6SYD_1 104 76.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]