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Parikh vectors
2ORR_1 8BQX_1 1MJB_1 Letter Amino acid
20 11 8 Q Glutamine
23 26 12 G Glycine
35 24 32 L Leucine
25 21 6 A Alanine
17 20 10 N Asparagine
30 25 18 E Glutamic acid
17 14 7 H Histidine
23 22 21 I Isoleucine
22 15 20 R Arginine
7 14 5 C Cysteine
12 8 6 M Methionine
23 25 12 P Proline
14 22 15 S Serine
24 33 17 T Threonine
20 23 9 V Valine
17 20 19 D Aspartic acid
18 9 12 F Phenylalanine
12 4 4 W Tryptophan
16 18 22 Y Tyrosine
14 33 23 K Lycine 

2ORR_1|Chain A|Nitric oxide synthase, inducible|Mus musculus (10090)
>8BQX_1|Chain A[auth x]|Methionine aminopeptidase 1|Saccharomyces cerevisiae (4932)
>1MJB_1|Chain A|Esa1 protein|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
2ORR , Knot 172 389 0.88 40 238 376 
MNPKSLTRGPRDKPTPLEELLPHAIEFINQYYGSFKEAKIEEHLARLEAVTKEIETTGTYQLTLDELIFATKMAWRNAPRCIGRIQWSNLQVFDARNCSTAQEMFQHICRHILYATNNGNIRSAITVFPQRSDGKHDFRLWNSQLIRYAGYQMPDGTIRGDAATLEFTQLCIDLGWKPRYGRFDVLPLVLQADGQDPEVFEIPPDLVLEVTMEHPKYEWFQELGLKWYALPAVANMLLEVGGLEFPACPFNGWYMGTEIGVRDFCDTQRYNILEEVGRRMGLETHTLASLWKDRAVTEINVAVLHSFQKQNVTIMDHHTASESFMKHMQNEYRARGGCPADWIWLVPPVSGSITPVFHQEMLNYVLSPFYYYQIEPWKTHIWQNEHHHH
8BQX , Knot 171 387 0.87 40 243 374 
MSTATTTVTTSDQASHPTKIYCSGLQCGRETSSQMKCPVCLKQGIVSIFCDTSCYENNYKAHKALHNAKDGLEGAYDPFPKFKYSGKVKASYPLTPRRYVPEDIPKPDWAANGLPVSEQRNDRLNNIPIYKKDQIKKIRKACMLGREVLDIAAAHVRPGITTDELDEIVHNETIKRGAYPSPLNYYNFPKSLCTSVNEVICHGVPDKTVLKEGDIVNLDVSLYYQGYHADLNETYYVGENISKEALNTTETSRECLKLAIKMCKPGTTFQELGDHIEKHATENKCSVVRTYCGHGVGEFFHCSPNIPHYAKNRTPGVMKPGMVFTIEPMINEGTWKDMTWPDDWTSTTQDGKLSAQFEHTLLVTEHGVEILTARNKKSPGGPRQRIK
1MJB , Knot 123 278 0.83 40 169 266 
MKEVARVRNLNRIIMGKYEIEPWYFSPYPIELTDEDFIYIDDFTLQYFGSKKQYERYRKKCTLRHPPGNEIYRDDYVSFFEIDGRKQRTWCRNLCLLSKLFLDHKTLYYDVDPFLFYCMTRRDELGHHLVGYFSKEKESADGYNVACILTLPQYQRMGYGKLLIEFSYELSKKENKVGSPQKPLSDLGLLSYRAYWSDTLITLLVEHQKEITIDEISSMTSMTTTDILHTAKTLNILRYYKGQHIIFLNEDILDRYNRLKAKKRRTIDPNRLIWKPPV

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 \(c\).

\(|P_{f(2ORR_1)}(2) \setminus P_{f(8BQX_1)}(2)|=90\), \(|P_{f(8BQX_1)}(2) \setminus P_{f(2ORR_1)}(2)|=95\). 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 1:10100100110001011001110110110000101001010001101011000100010001010011110011100110011010100101101000001001100100011010001010011011100001000101100011001100110101010110101001010111010010101111110101001011011101110101001000110011101011111101110111101110110110110011100100000001100110011100001101100011001011110010000101100001000110010000010110110111111110101011100011001101100001011000110000000
Pair \(Z_2\) Length of longest common subsequence
2ORR_1,8BQX_1 185 5
2ORR_1,1MJB_1 181 4
8BQX_1,1MJB_1 178 4

Newick tree

 
[
	2ORR_1:92.32,
	[
		1MJB_1:89,8BQX_1:89
	]:3.32
]

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 \((0.85)(0.8)(\frac{776 }{\log_{20} 776}-\frac{387}{\log_{20}387})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 2ORR_1 8BQX_1 136 135.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]