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Parikh vectors
4HXD_1 1OHH_1 6DAH_1 Letter Amino acid
0 2 0 C Cysteine
7 39 8 I Isoleucine
3 17 2 P Proline
7 26 12 T Threonine
2 49 11 A Alanine
2 13 5 N Asparagine
1 5 2 H Histidine
0 0 0 W Tryptophan
6 32 21 E Glutamic acid
1 10 10 M Methionine
1 16 2 Y Tyrosine
4 41 7 V Valine
5 29 17 D Aspartic acid
6 24 6 Q Glutamine
9 48 9 L Leucine
7 31 8 K Lycine
2 14 8 F Phenylalanine
3 33 5 S Serine
4 32 6 R Arginine
5 49 12 G Glycine 

4HXD_1|Chains A, C|Polyubiquitin-C|Homo sapiens (9606)
>1OHH_1|Chains A, B, C|ATP synthase subunit alpha, mitochondrial|Bos taurus (9913)
>6DAH_1|Chains A, B, C, D|Calmodulin-1|Homo sapiens (9606)
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 \(w=f(c)\)
4HXD , Knot 39 75 0.74 36 60 73 
MQIFVKTLTGKTITLEVEPSDTIENVKAKIQDKEGIPPDQQRLIFAGKQLEDGRTLSDYNIQKESTLHLVLRLRG
1OHH , Knot 205 510 0.83 38 229 476 
QKTGTAEVSSILEERILGADTSVDLEETGRVLSIGDGIARVHGLRNVQAEEMVEFSSGLKGMSLNLEPDNVGVVVFGNDKLIKEGDIVKRTGAIVDVPVGEELLGRVVDALGNAIDGKGPIGSKARRRVGLKAPGIIPRISVREPMQTGIKAVDSLVPIGRGQRELIIGDRQTGKTSIAIDTIINQKRFNDGTDEKKKLYCIYVAIGQKRSTVAQLVKRLTDADAMKYTIVVSATASDAAPLQYLAPYSGCSMGEYFRDNGKHALIIYDDLSKQAVAYRQMSLLLRRPPGREAYPGDVFYLHSRLLERAAKMNDAFGGGSLTALPVIETQAGDVSAYIPTNVISITDGQIFLETELFYKGIRPAINVGLSVSRVGSAAQTRAMKQVAGTMKLELAQYREVAAFAQFGSDLDAATQQLLSRGVRLTELLKQGQYSPMAIEEQVAVIYAGVRGYLDKLEPSKITKFENAFLSHVISQHQALLGKIRTDGKISEESDAKLKEIVTNFLAGFEA
6DAH , Knot 69 151 0.76 36 101 138 
GHMADQLTEEQIAEFKEAFSLFDKDGDGTITTKELGTVMRSLGQNPTEAELQDMINEVDADGNGTIDFPEFLTMMARKMKDTDSEEEIREAFRVFDKDGSGYISAAELRHVMTNLGEKLTDEEVDEMIREADIDGDGQVNYEEFVQMMTAK

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 \(c\).

\(|P_{f(4HXD_1)}(2) \setminus P_{f(1OHH_1)}(2)|=10\), \(|P_{f(1OHH_1)}(2) \setminus P_{f(4HXD_1)}(2)|=179\). 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 1:101110010100101010100010010101000011110000111110010010010000100000101110101
Pair \(Z_2\) Length of longest common subsequence
4HXD_1,1OHH_1 189 4
4HXD_1,6DAH_1 131 3
1OHH_1,6DAH_1 180 3

Newick tree

 
[
	1OHH_1:99.61,
	[
		4HXD_1:65.5,6DAH_1:65.5
	]:34.11
]

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 \((0.85)(0.8)(\frac{585 }{\log_{20} 585}-\frac{75}{\log_{20}75})=151.\)
Status Protein1 Protein2 d d1/2
Query variables 4HXD_1 1OHH_1 189 107
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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