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Parikh vectors
8TEF_1 2DMR_1 1MVZ_1 Letter Amino acid
20 88 0 G Glycine
40 33 2 I Isoleucine
16 25 0 M Methionine
22 57 8 T Threonine
7 18 0 W Tryptophan
0 4 14 C Cysteine
7 22 1 H Histidine
19 48 6 P Proline
28 30 2 Y Tyrosine
29 63 1 V Valine
25 87 2 A Alanine
10 36 4 R Arginine
35 22 0 N Asparagine
46 54 1 E Glutamic acid
56 59 1 L Leucine
55 39 3 K Lycine
26 32 3 F Phenylalanine
31 48 4 D Aspartic acid
16 19 3 Q Glutamine
32 39 7 S Serine 

8TEF_1|Chains A, B, C, D|Response regulator receiver protein|Flavobacterium johnsoniae UW101 (376686)
>2DMR_1|Chain A|DMSO REDUCTASE|Rhodobacter capsulatus (1061)
>1MVZ_1|Chain A|Bowman-Birk type protease inhibitor, (MSTI)|Medicago scutellata (36901)
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 \(w=f(c)\)
8TEF , Knot 207 520 0.83 38 238 490 
GSHMDKIRILWVDDEIDLLKPHILFLEKKNYEVTTSNNGLDAIALFEEENFDIVFLDENMPGMSGLETLSEMKEKKSAIPMIMITKSEEEYIMEEAIGSKIADYLIKPVNPNQILLSLKKNLDDSRLITEKTTLDYQKEFRKISMELAMVNSYEDWVELYKKLLFWELKLEDINDQAMIEILESQKVEANSQFGKYIERNYEDWFAPKADKPIQSHNLFKELVVPEIKKKDKPILFVVIDNLRYDQWKSFETVISNYYKLEKEVPYFSILPTATQYARNAIFSGLMPLDMEKQFPQYWKNDVEDGGKNLYEAEFLSAQIKRLGLNIKEDYFKITNYAGGKKLAENFKALKGNDLVTVVYNFVDMLSHAKTEMEVVKELASDDKAYRSLTLSWFKNSPLLEIIQQAQLLGFKLILTTDHGTINVKNPSKVVGDKNTSLNLRYKTGRSLTYEQKDVYVVKEPKTIGLPAINMSSSFIFAKNDFFLAYVNNYNHYVSYYKNTYQHGGISLEEMIIPFLVFNPK
2DMR , Knot 311 823 0.84 40 293 745 
MTKFSGNELRAELYRRAFLSYSVAPGALGMFGRSLLAKGARAEALANGTVMSGSHWGVFTATVENGRATAFTPWEKDPHPSPMLAGVLDSIYSPTRIKYPMVRREFLEKGVNADRSTRGNGDFVRVSWDQALDLVAAEVKRVEETYGPEGVFGGSYGWKSPGRLHNCTTLLRRMLTLAGGYVNGAGDYSTGAAQVIMPHVVGTLEVYEQQTAWPVLAENTEVMVFWAADPIKTSQIGWVIPEHGAYPGLEALKAKGTKVIVIDPVRTKTVEFFGAEHITPKPQTDVAIMLGMAHTLVAEDLYDKDFIANYTSGFDKFLPYLDGETDSTPKTAEWAEGISGVPAETIKELARLFESKRTMLAAGWSMQRMHHGEQAHWMLVTLASMLGQIGLPGGGFGLSYHYSGGGTPSTSGPALAGITDGGAATKGPEWLAASGASVIPVARVVDMLENPGAEFDFNGTRSKFPDVKMAYWVGGNPFVHHQDRNRMVKAWEKLETFVVHDFQWTPTARHADIVLPATTSYERNDIETIGDYSNTGILAMKKIVEPLYEARSDYDIFAAVAERLGKGAEFTEGKDEMGWIKSFYDDAAKQGKAAGVQMPAFDAFWAEGIVEFPVTDGADFVRYASFREDPLLNPLGTPTGLIEIYSKNIEKMGYDDCPAHPTWMEPLERLDGPGAKYPLHIAASHPFNRLHSQLNGTVLREGYAVQGHEPCLMHPDDAAARGIADGDVVRVHNDRGQILTGVKVTDAVMKGVIQIYEGGWYDPSDVTEPGTLDKYGDVNVLSADIGTSKLAQGNCGQTVLAEVEKYTGPAVTLTGFVAPKAAE
1MVZ , Knot 31 62 0.68 32 45 55 
TKSTTTACCDFCPCTRSIPPQCQCTDVREKCHSACKSCLCTRSFPPQCRCYDITDFCYPSCS

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 \(c\).

\(|P_{f(8TEF_1)}(2) \setminus P_{f(2DMR_1)}(2)|=38\), \(|P_{f(2DMR_1)}(2) \setminus P_{f(8TEF_1)}(2)|=93\). 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 1:1001001011110001011010111100000010000011011111000010111100011110110010010000011111110000000110011100110011011010011101000100001100000100000100101011110000011010001111010100100011101100001010001100100000011110100110000110011110100000111111100100001001001100000100011010111010001001110111110100011001000100110010010110101001110100001010001110011001011010011011001101100100010110011000010001010110001110110010111101110000101010010011100000101000010010000001011001001111110100011110001111010000001000000000111010011111111010
Pair \(Z_2\) Length of longest common subsequence
8TEF_1,2DMR_1 131 4
8TEF_1,1MVZ_1 245 3
2DMR_1,1MVZ_1 280 3

Newick tree

 
[
	1MVZ_1:14.10,
	[
		8TEF_1:65.5,2DMR_1:65.5
	]:81.60
]

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 \((0.85)(0.8)(\frac{1343 }{\log_{20} 1343}-\frac{520}{\log_{20}520})=210.\)
Status Protein1 Protein2 d d1/2
Query variables 8TEF_1 2DMR_1 265 217
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]