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Parikh vectors
2JHV_1 3MAM_1 8YOT_1 Letter Amino acid
7 10 6 R Arginine
3 12 5 N Asparagine
2 11 8 Q Glutamine
7 13 8 Y Tyrosine
5 10 4 F Phenylalanine
6 10 4 P Proline
11 7 12 S Serine
9 23 3 D Aspartic acid
1 0 2 C Cysteine
11 28 8 L Leucine
13 24 4 K Lycine
4 5 2 M Methionine
12 22 12 V Valine
7 24 10 A Alanine
9 26 4 E Glutamic acid
2 0 6 H Histidine
7 18 4 I Isoleucine
10 16 13 G Glycine
9 9 7 T Threonine
3 2 2 W Tryptophan 

2JHV_1|Chains A, B, C, D, E, F|RHO GDP-DISSOCIATION INHIBITOR 1|HOMO SAPIENS (9606)
>3MAM_1|Chain A|Osmoprotection protein (ProX)|Archaeoglobus fulgidus (2234)
>8YOT_1|Chains A, C|nanobody|Vicugna pacos (30538)
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 \(w=f(c)\)
2JHV , Knot 67 138 0.79 40 108 133 
AMVPNVVVTGLTLVCSSAPGPLELDLTGDLESFKKQSFVLKEGVEYRIKISFRVNREIVSGMKYIQHTYRKGVKIDKTDYMVGSYGPRAAAYEFLTPVEEAPKGMLARGSYSIKSRFTDDDKTDHLSWEWNLTIKKDW
3MAM , Knot 119 270 0.82 36 165 257 
ERVVIGSKPFNEQYILANMIAILLEENGYKAEVKEGLGGTLVNYEALKRNDIQLYVEYTGTAYNVILRKQPPELWDQQYIFDEVKKGLLEADGVVVAAKLGFRDDAALAVRADWAEENGVEKISDLAEFADQLVFGSDPEFASRPDGLPQIKKVYGFEFKEVKQMEPTLMYEAIKNKQVDVIPAYTTDSRVDLFNLKILEDDKGALPPYDAIIIVNGNTAKDEKLISVLKLLEDRIDTDTMRALNYQYDVEKKDAREIAMSFLKEQGLVK
8YOT , Knot 59 124 0.76 40 94 118 
QVQLQESGGGLVQAGGSLRLSCAASGSIFVYVPMGWYRQAPGKEREFVAAISLGSNTNYADSVKGRFTISRDNAKNTVYLQMNSLKPEDTAVYYCAVTGLRIYPFVYWGQGTQVTVSSHHHHHH

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 \(c\).

\(|P_{f(2JHV_1)}(2) \setminus P_{f(3MAM_1)}(2)|=52\), \(|P_{f(3MAM_1)}(2) \setminus P_{f(2JHV_1)}(2)|=109\). 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 1:111101110110110001111101010101001000011100110001010101000110110010000001101000001110011011100110110011011110100010001000000001010101010001
Pair \(Z_2\) Length of longest common subsequence
2JHV_1,3MAM_1 161 3
2JHV_1,8YOT_1 144 4
3MAM_1,8YOT_1 171 3

Newick tree

 
[
	3MAM_1:86.40,
	[
		2JHV_1:72,8YOT_1:72
	]:14.40
]

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 \((0.85)(0.8)(\frac{408 }{\log_{20} 408}-\frac{138}{\log_{20}138})=81.2\)
Status Protein1 Protein2 d d1/2
Query variables 2JHV_1 3MAM_1 100 75
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]