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Parikh vectors
5POK_1 6JWR_1 1FDG_1 Letter Amino acid
7 21 0 M Methionine
7 33 0 S Serine
11 15 1 A Alanine
4 89 0 N Asparagine
13 51 0 E Glutamic acid
10 20 0 H Histidine
6 77 0 K Lycine
14 23 0 R Arginine
13 52 0 D Aspartic acid
8 20 0 Q Glutamine
7 17 2 G Glycine
10 53 0 V Valine
5 74 0 I Isoleucine
6 23 0 F Phenylalanine
6 16 0 P Proline
1 13 2 C Cysteine
18 66 0 L Leucine
5 22 1 T Threonine
0 2 0 W Tryptophan
5 41 0 Y Tyrosine 

5POK_1|Chains A, B|Bromodomain-containing protein 1|Homo sapiens (9606)
>6JWR_1|Chains A, B|7,8-dihydro-6-hydroxymethylpterin pyrophosphokinase-dihydropteroate synthase|Plasmodium falciparum (5833)
>1FDG_1|Chains A, B|DNA (5'-D(*CP*GP*TP*AP*CP*G)-3')|synthetic construct (32630)
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 \(w=f(c)\)
5POK , Knot 74 156 0.79 38 120 151 
MHHHHHHSSGVDLGTENLYFQSMEQVAMELRLTELTRLLRSVLDQLQDKDPARIFAQPVSLKEVPDYLDHIKHPMDFATMRKRLEAQGYKNLHEFEEDFDLIIDNCMKYNARDTVFYRAAVRLRDQGGVVLRQARREVDSIGLEEASGMHLPERPA
6JWR , Knot 274 728 0.82 40 256 634 
METIQELILSEENKTNIAVLNLGTNDRRNAVLILETALHLVEKYLGKIINTSYLYETVPEYIVLDKKESCEKINKDCRIYDVNYINELMQNLEESKYEENKELIDKCEEYETFLKNGKVDNSILKEVNVENYLLECNNIIVKNDEIMKNNLSKYKDKYYTSYFYNLTVVVKTFVNDPLSMLVVIKYIEELMKRENVKEKEKFENRIIDIDILFFNDFTIFMKNIKLEKNMIYKILSKYIHLERDIKNGNDNMSKVNMDKDINLNNNNNIKKKNNNDIDCDCVDQKMNNHVNNKNYINSFRDPQEIINNMVDNIEFLSIPHVYTTHRYSILLCLNDMIPEYKHNVLNNTIRCLYNKYVSRMKEQYNINIKENNKRIYVLKDRISYLKEKTNIVGILNVNYDSFSDGGIFVEPKRAVQRMFEMINEGASVIDIGGESSAPFVIPNPKISERDLVVPVLQLFQKEWNDIKNKIVKCDAKPIISIDTINYNVFKECVDNDLVDILNDISACTNNPEIIKLLKKKNKFYSVVLMHKRGNPHTMDKLTNYDNLVYDIKNYLEQRLNFLVLNGIPRYRILFDIGLGFAKKHDQSIKLLQNIHVYDEYPLFIGYSRKRFIAHCMNDQNVVINTQQKLHDEQQNENKNIVDKSHNWMFQMNYMRKDKDQLLYQKNICGGLAIASYSYYKKVDLIRVHDVLETKSVLDVLTKIDQVKDPNSSSVDKLAAALEHHHHHH
1FDG , Knot 5 6 0.49 8 4 4 
CGTACG

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 \(c\).

\(|P_{f(5POK_1)}(2) \setminus P_{f(6JWR_1)}(2)|=33\), \(|P_{f(6JWR_1)}(2) \setminus P_{f(5POK_1)}(2)|=169\). 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 1:100000000110110001010010011101010010011001100100001101110110100110010010011011010001010100010010001011100010001000110011101000111110010001001110010110110011
Pair \(Z_2\) Length of longest common subsequence
5POK_1,6JWR_1 202 6
5POK_1,1FDG_1 122 2
6JWR_1,1FDG_1 252 2

Newick tree

 
[
	6JWR_1:12.06,
	[
		5POK_1:61,1FDG_1:61
	]:66.06
]

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 \((0.85)(0.8)(\frac{884 }{\log_{20} 884}-\frac{156}{\log_{20}156})=202.\)
Status Protein1 Protein2 d d1/2
Query variables 5POK_1 6JWR_1 246 149
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]