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Parikh vectors
4JRW_1 6OSM_1 6HJZ_1 Letter Amino acid
4 3 12 N Asparagine
0 4 4 M Methionine
8 4 22 S Serine
1 0 5 W Tryptophan
6 19 32 V Valine
2 0 18 R Arginine
8 6 19 D Aspartic acid
0 0 1 C Cysteine
5 18 16 E Glutamic acid
11 18 8 G Glycine
3 4 15 Y Tyrosine
1 4 31 L Leucine
8 19 25 A Alanine
0 6 7 Q Glutamine
1 1 3 H Histidine
5 2 28 I Isoleucine
10 15 10 K Lycine
3 2 24 F Phenylalanine
2 5 18 P Proline
9 10 19 T Threonine 

4JRW_1|Chains A, B|Collagenase|Clostridium histolyticum (1498)
>6OSM_1|Chains A, B, C, D, E, F, G, H, I, J|Alpha-synuclein|Homo sapiens (9606)
>6HJZ_1|Chains A, B, C, D, E, F, G, H, I, J|Proton-gated ion channel|Gloeobacter violaceus (strain PCC 7421) (251221)
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 \(w=f(c)\)
4JRW , Knot 46 87 0.78 34 72 83 
GSAPIAKVTGPSTGAVGRNIEFSGKDSKDEDGKIVSYDWDFGDGATSRGKNSVHAYKKAGTYNVTLKVTDDKGATATESFTIEIKNE
6OSM , Knot 59 140 0.69 34 86 121 
MDVFMKGLSKAKEGVVAAAEKTKQGVAEAAGKTKEGVLYVGSKTKEGVVHGVATVAEKTKEQVTNVGGAVVTGVTAVAQKTVEGAGSIAAATGFVKKDQLGKNEEGAPQEGILEDMPVDPDNEAYEMPSEEGYQDYEPEA
6HJZ , Knot 137 317 0.83 40 187 301 
GQDMVSPPPPIADEPLTVNTGIYLIECYSLDDKAETFKVNAFLSLSWKDRRLAFDPVRSGVRVKTYEPEAIWIPEIRFVNVENARDADVVDISVSPDGTVQYLERFSARVLSPLDFRRYPFDSQTLHIYLIVRSVDTRNIVLAVDLEKVGKNDDVFLTGWDIESFTAVVKPANFALEDRLESKLDYQLRISRQYFSYIPNIILPMLFILFISWTAFWSTSYEANVTLVVSTLIAHIAFNILVETNLPKTPYMTYTGAIIFMIYLFYFVAVIEVTVQHYLKVESQPARAASITRASRIAFPVVFLLANIILAFLFFGF

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 \(c\).

\(|P_{f(4JRW_1)}(2) \setminus P_{f(6OSM_1)}(2)|=47\), \(|P_{f(6OSM_1)}(2) \setminus P_{f(4JRW_1)}(2)|=61\). 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 1:101111010110011110010101000000010110001011011000100010100011000101010000110100010101000
Pair \(Z_2\) Length of longest common subsequence
4JRW_1,6OSM_1 108 3
4JRW_1,6HJZ_1 169 4
6OSM_1,6HJZ_1 175 5

Newick tree

 
[
	6HJZ_1:94.29,
	[
		4JRW_1:54,6OSM_1:54
	]:40.29
]

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 \((0.85)(0.8)(\frac{227 }{\log_{20} 227}-\frac{87}{\log_{20}87})=45.5\)
Status Protein1 Protein2 d d1/2
Query variables 4JRW_1 6OSM_1 53 44.5
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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