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Parikh vectors
5PSF_1 7FYZ_1 4YIY_1 Letter Amino acid
4 4 31 N Asparagine
13 11 30 D Aspartic acid
1 2 15 C Cysteine
18 5 59 L Leucine
7 10 55 S Serine
10 17 48 V Valine
6 6 28 F Phenylalanine
5 2 13 Y Tyrosine
11 6 49 A Alanine
14 6 48 R Arginine
8 2 28 Q Glutamine
5 8 35 I Isoleucine
6 14 17 K Lycine
7 5 15 M Methionine
5 12 40 T Threonine
13 9 40 E Glutamic acid
7 11 64 G Glycine
10 2 27 H Histidine
6 1 37 P Proline
0 2 1 W Tryptophan 

5PSF_1|Chains A, B|Bromodomain-containing protein 1|Homo sapiens (9606)
>7FYZ_1|Chain A|Fatty acid-binding protein, adipocyte|Homo sapiens (9606)
>4YIY_1|Chains A, B|kRNA Editing A6 Specific Protein|Trypanosoma brucei brucei (strain 927/4 GUTat10.1) (185431)
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 \(w=f(c)\)
5PSF , Knot 74 156 0.79 38 120 151 
MHHHHHHSSGVDLGTENLYFQSMEQVAMELRLTELTRLLRSVLDQLQDKDPARIFAQPVSLKEVPDYLDHIKHPMDFATMRKRLEAQGYKNLHEFEEDFDLIIDNCMKYNARDTVFYRAAVRLRDQGGVVLRQARREVDSIGLEEASGMHLPERPA
7FYZ , Knot 69 135 0.83 40 104 129 
GSHMCDAFVGTWKLVSSENFDDYMKEVGVGFATRKVAGMAKPNMIISVNGDVITIKSESTFKNTEISFILGQEFDEVTADDRKVKSTITLDGGVLVHVQKWDGKSTTIKRKREDDKLVVECVMKGVTSTRVYERA
4YIY , Knot 259 680 0.82 40 277 626 
MHHHHHHSSGVDLGTENLYFQSRGGRGGGRGHSHWHNNAPMERGGHMPSNNEGAAISSGSTGFSPLMDFFHSVEGRNYGELRSLTNETYQISENVRCTFLSIQSDPFAPGSQVRLVCPCTFSLEKVLQTTDLAAANPCRRVAAEDFILRSFHAGYRNGIPRRTSGAVQVLRPSQHVLERSTVGLVKAHQQKSGMQAEIEIFARVKLPGHGRRIDGHGAIDIFYNELVPLLEQCVVGLNEEDLHQHVICVHDQEELRSNLLGAGYVAFVANGAILPRDAGNSDKPLRDNAVPFQSPKSLECSFTLPHSGKTITGMGLPPGLTLIAGGGFHGKSTLLRALEVGIYNHVPDDGRTYVVVDPTAVKIRAEDRRSVHGVDISPFINNLPFGKTTNFFVTADASGSTSQAANIMEALELGSQLLLLDEDTCATNLMYRDALMQMLVPRAQEPITPFVERVADLSQNHGVSSIMVIGGSGQYFPQARVVLVMNAYQISDCTKEAKEIASNSSLPALNPPGDTASVFIPDVNRCFDPDGSFTTVRRRRGREGTKVSGIGTESIRFSEETIDLSMVEQIVEEGQVNAIAQCLALLYDGEPRIVPEMTTKGGALTQLPSPGGVCEIQRGKFNSNFSSMIAGCCSHQHDKRLELRTPSCYLPRGFTSATRHIEIGAALNRLRTLRTVTAKR

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 \(c\).

\(|P_{f(5PSF_1)}(2) \setminus P_{f(7FYZ_1)}(2)|=87\), \(|P_{f(7FYZ_1)}(2) \setminus P_{f(5PSF_1)}(2)|=71\). 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
5PSF_1,7FYZ_1 158 3
5PSF_1,4YIY_1 213 22
7FYZ_1,4YIY_1 215 4

Newick tree

 
[
	4YIY_1:11.82,
	[
		5PSF_1:79,7FYZ_1:79
	]:35.82
]

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 \((0.85)(0.8)(\frac{291 }{\log_{20} 291}-\frac{135}{\log_{20}135})=48.4\)
Status Protein1 Protein2 d d1/2
Query variables 5PSF_1 7FYZ_1 63 59.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]