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Parikh vectors
4YUM_1 1HGB_1 7LHQ_1 Letter Amino acid
9 21 5 A Alanine
10 0 2 I Isoleucine
14 11 6 K Lycine
1 1 1 W Tryptophan
9 4 3 N Asparagine
4 10 1 H Histidine
5 2 4 M Methionine
11 9 4 T Threonine
8 11 10 S Serine
6 3 2 R Arginine
3 1 6 Q Glutamine
7 18 15 L Leucine
15 7 1 F Phenylalanine
6 7 0 P Proline
2 3 0 Y Tyrosine
9 13 9 V Valine
7 8 5 D Aspartic acid
4 1 3 C Cysteine
12 4 5 E Glutamic acid
23 7 2 G Glycine 

4YUM_1|Chain A|Peptidyl-prolyl cis-trans isomerase A|Homo sapiens (9606)
>1HGB_1|Chains A, C|HEMOGLOBIN (AQUO MET) (ALPHA CHAIN)|Homo sapiens (9606)
>7LHQ_1|Chain A|Non-structural protein 7|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
4YUM , Knot 82 165 0.84 40 129 161 
MVNPTVFFDIAVDGEPLGRVSFELFADKVPKTAENFRALSTGEKGFGYKGSCFHRIIPGFMCQGGDFTRHNGTGGKSIYGEKFEDENFILKHTGPGILSMANAGPNTNGSQFFICTAKTEWLDGKHVVFGKVKEGMNIVEAMERFGSRNGKTSKKITIADCGQLE
1HGB , Knot 69 141 0.80 38 99 137 
VLSPADKTNVKAAWGKVGAHAGEYGAEALERMFLSFPTTKTYFPHFDLSHGSAQVKGHGKKVADALTNAVAHVDDMPNALSALSDLHAHKLRVDPVNFKLLSHCLLVTLAAHLPAEFTPAVHASLDKFLASVSTVLTSKYR
7LHQ , Knot 44 84 0.77 36 67 77 
GSKMSDVKCTSVVLLSVLQQLRVESSSKLWAQCVQLHNDILLAKDTTEAFEKMVSLLSVLLSMQGAVDINKLCEEMLDNRATLQ

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 \(c\).

\(|P_{f(4YUM_1)}(2) \setminus P_{f(1HGB_1)}(2)|=91\), \(|P_{f(1HGB_1)}(2) \setminus P_{f(4YUM_1)}(2)|=61\). 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 1:110101110111010111010101110011001001011001001110010010011111100110100001011001010010000111000111110110111000100111001000110100111101001101101100110001000001011001010
Pair \(Z_2\) Length of longest common subsequence
4YUM_1,1HGB_1 152 3
4YUM_1,7LHQ_1 144 3
1HGB_1,7LHQ_1 118 3

Newick tree

 
[
	4YUM_1:78.39,
	[
		7LHQ_1:59,1HGB_1:59
	]:19.39
]

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 \((0.85)(0.8)(\frac{306 }{\log_{20} 306}-\frac{141}{\log_{20}141})=50.8\)
Status Protein1 Protein2 d d1/2
Query variables 4YUM_1 1HGB_1 67 60.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]