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\begin{document}

\title{Comprehensive Folding Variations for Protein Folding}



\author[1]{Jiaan Yang}%
\author[1]{Wen Xiang}%
\author[2]{Xiao Fei Zhao}%
\author[3]{Gang Wu}%
\author[4]{Shi Tong}%
\author[5]{Qiyue Hu}%
\author[5]{Hu Ge}%
\author[5]{Qianshan Qin}%
\author[5]{Xinshen Jin}%
\author[5]{Lianshan Zhang}%
\author[1]{Peng Zhang}%
\affil[1]{Chinese Academy of Sciences Shenzhen Institutes of Advanced Technology}%
\affil[2]{Micro Biotech Ltd Shanghai 200123 China}%
\affil[3]{Huazhong University of Science and Technology Tongji Medical College}%
\affil[4]{Shenzhen Hua Ying Kang Gene Technology Co Ltd Shenzhen Guangdong 518057 China}%
\affil[5]{Shanghai Hengrui Pharmaceutical CoLtd Shanghai 200245 China}%


\vspace{-1em}



  
  \date{May 18, 2024}


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\selectlanguage{english}
\begin{abstract}
The revelation of protein folding is a challenging subject in both
discovery and description. Except acquirement of accurate 3D structure
for protein stable state, another big hurdle is how to discover
structural flexibility for protein innate character. Even if a huge
number of flexible conformations are known, difficulty is how to
describe these conformations. A novel approach, protein structure
fingerprint, has been developed to expose the comprehensive local
folding variations, and then construct folding conformations for entire
protein. The backbone of 5 amino acid residues was identified as a
universal folden, and then a set of Protein Folding Shape Code (PFSC)
was derived for completely covering folding space in alphabetic
description. Sequentially, a database was created to collect all
possible folding shapes of local folding variations for all permutation
of 5 amino acids. Successively, Protein Folding Variation Matrix (PFVM)
assembled all possible local folding variations along sequence for a
protein, which possesses several prominent features. First, it showed
the fluctuation with certain folding patterns along sequence which
revealed how the protein folding was related the order of amino acids in
sequence. Second, all folding variations for an entire protein can be
simultaneously apprehended at a glance within PFVM. Third, all
conformations can be determined by local folding variations from PFVM,
so total number of conformations is no longer ambiguous for any protein.
Finally, the most possible folding conformation and its 3D structure can
be acquired according PFVM for protein~structure~prediction. Therefore,
the protein structure fingerprint approach provides a significant means
for investigation of protein folding problem.%
\end{abstract}%



\sloppy


\textbf{Comprehensive Folding Variations for Protein Folding}

Jiaan Yang\textsuperscript{*, \#, 1,2}, Wen Xiang
Cheng\textsuperscript{\#, 1}, Xiao Fei Zhao\textsuperscript{2}, Gang
Wu\textsuperscript{3}, Shi Tong Sheng\textsuperscript{4},

Qiyue Hu\textsuperscript{5}, Hu Ge\textsuperscript{5}, Qianshan
Qin\textsuperscript{5}, Xinshen Jin\textsuperscript{5}, Lianshan
Zhang\textsuperscript{5}, Peng Zhang\textsuperscript{1}

\textsuperscript{1} Shenzhen Institutes of Advanced Technology, Chinese
Academy of Sciences, Shenzhen, Guangdong, 518055, China

\textsuperscript{2} Micro Biotech, Ltd., Shanghai, 200123, China

\textsuperscript{3} School of Basic Medicine, Tongji Medical College,
Huazhong University of Science and Technology, 430030 Wuhan, China

\textsuperscript{4} Shenzhen Hua Ying Kang Gene Technology Co., Ltd,
Shenzhen, Guangdong, 518057, China

\textsuperscript{5} Shanghai Hengrui Pharmaceutical Co.Ltd., Shanghai
200245, China

\textsuperscript{\#} Contributed equally to this work

* Corresponding author: jiaanyang@yahoo.com

Keywords: protein folding, protein conformation, folding conformation,
protein~structure~prediction, intrinsically disordered protein,
alphabetic description

\textbf{ABSTRACT}

The revelation of protein folding is a challenging subject in both
discovery and description. Except acquirement of accurate 3D structure
for protein stable state, another big hurdle is how to discover
structural flexibility for protein innate character. Even if a huge
number of flexible conformations are known, difficulty is how to
describe these conformations. A novel approach, protein structure
fingerprint, has been developed to expose the comprehensive local
folding variations, and then construct folding conformations for entire
protein. The backbone of 5 amino acid residues was identified as a
universal folden, and then a set of Protein Folding Shape Code (PFSC)
was derived for completely covering folding space in alphabetic
description. Sequentially, a database was created to collect all
possible folding shapes of local folding variations for all permutation
of 5 amino acids. Successively, Protein Folding Variation Matrix (PFVM)
assembled all possible local folding variations along sequence for a
protein, which possesses several prominent features. First, it showed
the fluctuation with certain folding patterns along sequence which
revealed how the protein folding was related the order of amino acids in
sequence. Second, all folding variations for an entire protein can be
simultaneously apprehended at a glance within PFVM. Third, all
conformations can be determined by local folding variations from PFVM,
so total number of conformations is no longer ambiguous for any protein.
Finally, the most possible folding conformation and its 3D structure can
be acquired according PFVM for protein~structure~prediction. Therefore,
the protein structure fingerprint approach provides a significant means
for investigation of protein folding problem.

\textbf{INTRODUCTION}

Protein folding is one of the challenging subjects in science,11Science
1 July:Vol. 309 no. 5731 pp. 78-102 (2005).,22Dill K A, Maccallum J L.
The protein-folding problem, 50 years on.{[}J{]}. Science,
338(6110):1042 (2012). which particularly has attracted much attention
since recent progress by AlphaFold. With artificial intelligence (AI)
approach, AlphaFold made a significant breakthrough to accurately
predict 3D structures based on protein sequence.33~J. Jumper, etc.
''Highly accurate protein structure prediction with AlphaFold''. Nature.
596 (7873): 583--589 (2021).~,44Callaway, Ewen.~'''It will change
everything': DeepMind's AI makes gigantic leap in solving protein
structures''.~Nature.~588~(7837): 203--204 (2020).~ However, the protein
folding problem has not been thoroughly resolved yet because protein is
not a static structure.55~Stephen Curry,~No, DeepMind has not solved
protein folding, Reciprocal Space (blog), 2 December (2020).,66Balls,
Phillip.~''Behind the screens of AlphaFold''.~Chemistry World,
(2020).~The intrinsically disordered protein~(IDP) has already
discovered that many proteins lacked a fixed three-dimensional
structure, and many protein functions were accomplished with~ensemble~of
flexible conformations.77Robin van der Lee and etc, Classification of
intrinsically disordered regions and proteins.{[}J{]}. Chemical Reviews,
2014, 114(13):6589.,88Dunker AK, Lawson JD, Brown CJ, Williams RM,
Romero P, Oh JS, Oldfield CJ, Campen AM, Ratliff CM, Hipps KW, Ausio J,
Nissen MS, Reeves R, Kang C, Kissinger CR, Bailey RW, Griswold MD, Chiu
W, Garner EC, Obradovic Z (2001). ''Intrinsically disordered
protein''.~Journal of Molecular Graphics \& Modelling.~19~(1):
26--59.,99Dyson HJ, Wright PE (March 2005). ''Intrinsically unstructured
proteins and their functions''.~Nature Reviews Molecular Cell
Biology.~6~(3): 197--208,1010Dunker AK, Silman I, Uversky VN, Sussman JL
(December 2008). ''Function and structure of inherently disordered
proteins''.~Current Opinion in Structural Biology.~18~(6): 756--64.~
Thus, except for accuracy in structure, biologists also want to know how
many different ways the protein will fold into, why protein are in such
folds and what biologic functions are impacted by folding. To date, it
is well known that the protein folding patterns are primarily decided by
global multiple weak interactions of protein itself, such as hydrogen
bond, disulfide bond, van der Waals force, electrostatic interactions
and hydrophobic interactions, etc. Also, it is influenced by environment
factors, such as protein-protein interactions, ligands, ions,
solvent,pH,temperature and chaperones, etc. Under constraints, a
protein can still fold into various conformations between random coil
and native state, or undergo reversible folding process between disorder
and order transitions. In 1957, Francis Crick indicated that protein
folding was simply a function of the order of amino acids.1111Cobb M. 60
years ago, Francis Crick changed the logic of biology{[}J{]}. Plos
Biology, 15(9):e2003243, (2017). It is true that different order of
amino acids or replacement of residue in sequence may cause the change
in folding conformation. In 1969, Cyrus Levinthal indicated that protein
may have an astronomical number of local minima in conformational
space,1212Levinthal, C. How to Fold Graciously. In Mossbauer~
\selectlanguage{english}Spectroscopy in Biological Systems, pp 22-24, Allerton House,
Monticello, IL (1969). and further pointed out to understand the
relationship from sequence to protein folding was a challenging problem.
The basic task includes how to obtain all possible folding
conformations, how to present these folding conformations with an
astronomical number and how to acquire the most possible conformation in
stable state and its 3D structure.

In~spite~of the lack of systematical approach, some protein
conformations are still known. The protein conformations may be obtained
by protein 3D structures which are experimental measurement data or
results of computational approaches. Experimental measurements, such as
X-ray crystallography, Nuclear Magnetic Resonance~(NMR) and Transmission
Electron Cryomicroscopy~(CryoTEM) etc., may accurately determine atomic
coordinates of protein 3D structures. However, they only provided the
limited folding conformations for protein stable states under specific
conditions. Anyhow, the results from experiments are snapshots of
protein structures which provide significant folding information, but
they couldn't cover the enormous conformational space. Also, the
progress of experimental measurements for protein 3D structures cannot
keep up with the pace of rapid increase of knowledge of protein
sequences as a huge number of protein sequences are determined by
genetic code. To date, over 35,000,000 gene codes are available in
National Center for Biotechnology Information (NCBI)
database,1313https://www.ncbi.nlm.nih.gov/ and over 225,000,000 protein
sequences in Universal Protein Resource (UniProt)
database.1414http://www.uniprot.org/ So far, merely about 187,000 of 3D
structures are available in Protein Data Bank
(PDB).1515https://www.rcsb.org/ In other words, less than 1\% of total
protein sequences have the known protein 3D structures. Therefore, on
the other hand, the development of computational approaches becomes an
important methodology to predict the protein 3D structures. The effort
of protein structure prediction, however, is primarily focusing on to
achieve structures with thermodynamic stability, not multiple states for
various folding conformations. In view of protein structure flexibility,
many databases have cumulated information about protein or sequence
regions involving intrinsically disordered protein~(IDP).1616Lazar, T.,
Mart\selectlanguage{polish}ı\selectlanguage{english}nez-P\selectlanguage{ngerman}érez, E., Quaglia, F., Hatos, A., Chemes, L.B., Iserte, J.A.,
Méndez, N.A., Garrone, N.A., Saldaño, T.E., Marchetti, J., Velez Rueda,
A.J., Bernadó, P., Blackledge, M., Cordeiro, T.N., Fagerberg, E.,
Forman-Kay, J.D., Fornasari, M.S., Gibson, T.J., Gomes, G-N.W.,
Gradinaru, C.C., Head-Gordon, T., Ringkjøbing Jensen, M., Lemke, E.A.,
Longhi, S., Marino-Buslje, C., Minervini, G., Mittag, T., Monzon, A.M.,
Pappu, R.V., Parisi, G., Ricard-Blum, S., Ruff, K.M., Salladini, E.,
Skepö, M., Svergun, D., Vallet, S.D., Varadi, M., Tompa, P., Tosatto,
S.C.E., Piovesan D.\selectlanguage{english}, PED in 2021: a major update of the Protein
Ensemble Database for intrinsically disordered proteins\selectlanguage{english},Nucleic Acids
Research, Volume 49, Issue D1, (2021) D404--D411,1717Damiano
Piovesan,~Marco Necci,~Nahuel Escobedo,~Alexander Miguel Monzon,~Andras
Hatos~\ldots{}Nucleic Acids Research, Volume 49, Issue D1, 8 January
2021, Pages D361--D367,1818Fukuchi, Satoshi et al. ``IDEAL:
Intrinsically Disordered proteins with Extensive Annotations and
Literature.''~Nucleic acids research~vol. 40, Database issue (2012):
D507-11.,1919Federica Quaglia,~Balint Meszaros,~Edoardo
Salladini,~Andras Hatos,~Rita Pancsa~\ldots{} DisProt in 2022: improved
quality and accessibility of~protein~intrinsic~disorder~annotation
Nucleic Acids Research~, Volume 50, Issue D1, 7 January 2022, Pages
D480--D487,2020Damiano Piovesan, Marco Necci, Nahuel Escobedo, Alexander
Miguel Monzon, Andras Hatos, Ivan Mi\selectlanguage{polish}č\selectlanguage{english}eti\selectlanguage{polish}ć, \selectlanguage{english}Federica Quaglia, Lisanna
Paladin, Pathmanaban Ramasamy, Zsuzsanna Doszt\selectlanguage{ngerman}ányi, Wim F Vranken,
Norman~E Davey, Gustavo Parisi, Monika Fuxreiter, Silvio~C E Tosatto,
MobiDB: intrinsically disordered proteins in 2021,~\emph{Nucleic Acids
Research} , Volume 49, Issue D1, 8 January 2021, Pages D361--D367 The
definitions of IDP are based on annotations of experimental data coming
mainly from Nuclear Magnetic Resonance (NMR), Small-angle X-ray
Scattering (SAXS) measurements and Molecular Dynamics (MD) simulations.
However, an optimal approach for protein folding should obtain all
possible folding conformations, expose folding difference between
regions within a protein or between different proteins, including
mutation or differentiation. Also, the most possible conformation and 3D
structure should be extracted from a massive number of conformations.

Here, the protein structure fingerprint as novel approach to reveals the
protein folding variations as well as the most possible conformation. A
folden of element of 5 amino acid residues is firstly defined to probe
the attribute of local folds, and then the local folds are extended to
entire protein system to discover all possible folding conformations.
First, a folden with 5 points connection as ball-and-stick is initiative
model and make mathematical derivation. Without biological structure
constrain, all folds are equivalently around each join point with
topological uniformity, and all possible folds in geometric space form a
complete and continuous aggregation. Second, the continuous aggregation
of folding description is simplified by partition of space to reduce
variable dimensions, and is applied to protein biological space. Then a
set of 27 folding shapes are obtained which is able completely to cover
various folding patterns for 5 successive amino acid residues. Third,
with alphabetic description, these 27 folding shapes are represented by
using 26 letters and ``\$'' symbol. Thus, the topological model
mathematically established the foundation to describe the protein
backbone folding. As a set of 27 letters is applied to protein systems,
it is called as the Protein Folding Shape Code (PFSC),2121Yang J,
Comprehensive description of protein structures using protein folding
shape code. Proteins;71.3:1497-1518 (2008). which essentially represent
the folding shapes of 5 amino acid residues.

For protein with known 3D structure, its complete folding conformation
can be described by PFSC string. The folding shape of any set of 5
successive amino acid residues is identified by a PFSC letter according
the given coordinates of alpha carbon atoms. Along sequence from
N-terminus to C-terminus, the conformation can be described by a string
of PFSC letters. As one PFSC letter represents a folding shape of 5
successive amino acid residues, and two adjacent PFSC letters actually
share partial of folding shape overlap of 4 amino acid residues. Thus
any protein folding conformations can be completely described by a PFSC
string, covering regular secondary structure fragments as well as
irregular tertiary structure fragments.

For protein without given 3D structure, the comprehensive folding
conformations for a protein are able to be exposed by local folding
variations. In order to achieve this goal, all of possible permutations
for 5 amino acids as well as all possible local folding shapes for each
5 amino acids are needed to well know. There are total 3,200,000
permutations for 5 amino acids based on 20 standard amino acids. For the
permutations of 5 amino acids available in PDB, all folds have been
first primarily collected. Then, for the permutations of 5 amino acids
not available in PDB, their 3D structures were calculated by
molecular~dynamics~simulations, and the folding shapes were obtained.
Consequently, a new database 5AAPFSC, where the folding shapes of 5
amino acids are described by PFSC, is created to assemble all possible
folding shapes for each permutations of 5 amino acid residues. A set of
5 amino acids may have one or more than one PFSC letters, but no more
than 27 PFSC assignments. Each set of 5 amino acids may have different
folding patterns and different number of PFSC letters. Therefore,
according sequence only, all the possible folding shapes for each
successive 5 amino acids from N-terminus to C-terminus can be thoroughly
represented by continue sets of PSFC letters. The local folding
variations are displayed in Protein Folding Variation Matrix
(PFVM)2222Yang J. Protein Structure Fingerprint Technology. J Bioinform,
Genomics, Proteomics X: 3(2): 1036, (2008). which the protein sequence
is listed horizontally, and all folding shapes in the PFSC letters for
each 5 successive amino acids are displayed vertically. The PFVM
provides rich information to promote protein folding investigation.
First, for a protein, the comprehensive local folding variations along
sequence are simultaneously exhibited by PFVM. Second, the local folding
variations are fluctuated with the folding pattern and number. Third,
all possible conformations with an astronomical number for a protein can
be assembled with various combinations of local folding variations, and
the most possible conformation and 3D structure can be easily
determined. Finally, the protein structure fingerprint produces the
ensemble~of conformations to probe the protein structures as well as the
application in biological drug design and disease research.2323Yang J \&
Lee WH, Protein Structure Alphabetic Alignment, Protein Structure,
Edited by Eshel Faraggi, InTech Publishers, (ISBN 978-953-51-0555-8),
133-156 (2012).,2424Yang J, Wu G, From Sequence to Protein Folding
Variations. Biomedical Journal of Scientific \& Technical Research,
(2019).,2525Yang J, Zhang P, Cheng W X, et al. Exposing Structural
Variations in SARS-CoV-2 Evolution. Scientifc Reports, 11:22042, (2021).
Thus, the protein structure fingerprint provided a signification
foundation for the solution of protein folding and applications.

\textbf{METHODS}

\textbf{Protein Folding Shape Code (PFSC).}

With protein folding fingerprint, the PFSC alphabetic string can provide
a complete description for protein conformation. Mathematically, 5
points with successive connection in geometric space was firstly
considered as a topological folding model. With derivation, the initial
higher dimensions of topological folding space were reduced and the
continuous space was partitioned, and then a set of 27 folding shapes
was obtained which are able completely to cover various folding patterns
for 5 points in sequential connection. These folding shapes can be
representing with 27 letters including ``\$'' symbol as a digitized
expression. However, for biological protein, a set of 5 successive amino
acid residues may not actually have all 27 folding shapes due to
structural constrain. With alphabetical expression, the 27 folding
shapes for 5 successive amino acid residues are called by Protein
Folding Shape Code (PFSC) and displayed in the cubic of Figure 1. With
integration feature, any PFSC letter in the cubic has partial folding
similarity with its surrounding neighboring letters, and then a 27 PFSC
may be transformed from one to each other by neighboring similarity. For
example, the letter ``A'' represents a typical alpha-helix, and ``H, D,
V, Y, J and P'' around ``A'' respectively has partial alpha-helical
character in folds. In the same column with ``A'', ``H'' is an extensive
helix fold and ``D'' is a compressive helix fold respectively. Nearby
``A'', ``V, Y, J or P'' has partial helical fold in N- or C-terminus
respectively. The letter ``B'' represents a typical beta-strand, and
``E, G, V, J, M and S'' around ``B'' respectively has partial
beta-strand in folds. Other remaining letters relate to irregular folds
for tertiary folds, and also have partial similarity with neighboring
letter in the cubic. Briefly, none of PFSC is isolated, and all 27 PFSC
letters are formed into a meaningful ensemble with structural relevance.
The blue arrows in Figure 1 indicated that the folding conformation for
a protein with known 3D structure is able to be completely described by
a PFSC string without gap. The PFSC letter is assigned to the folding
shape of each 5 consecutive residues from N-terminus to C- terminus, and
two PFSC letters next each other share folding shape of four amino
acids. In summary, for any protein with given 3D structure, the PFSC
alphabetical string is able to completely describe the folding
conformation, covering secondary structure fragments as well as tertiary
structure fragments.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

Figure 1. Protein folding shape code (PFSC) and protein folding
variation matrix (PFVM). The green arrows indicated the process to
construct 5AAPFSC database, which contained all folding shapes in PFSC
letters for 3,200,000 of permutations of 5 amino acids. The blue arrows
indicated the process how to obtain the PFSC string from a protein with
known 3D structure. The red arrows indicated the process how to obtain
the PFVM from a protein sequence.

\textbf{Protein Folding Variation Matrix (PFVM).}

With protein folding fingerprint, the PFVM assembles the local folding
variations along sequence, and it can construct an astronomical number
of conformations while define the most possible conformation. Firstly, a
database, which collects all possible folding shapes in PFSC letters, is
created. Based on 20 of amino acids, there are 3,200,000 of permutations
for 5 amino acids. All folding shapes for each permutations of 5 amino
acids are collected from database or calculations. The folding shapes
for most of permutations of 5 amino acids were firstly collected from 3D
structural data in PDB. For the permutations of 5 amino acids do not
exist in PDB, the folding shapes were computed by molecular dynamics
simulation method and stored into a complementary database. Then, all
folding shapes for 3,200,000 of permutations of 5 amino acids were
converted into the PFSC alphabetic letters, and stored into a database
named as 5AAPFSC. The procedure is indicated by green arrows in Figure
1. Actually, most of 5 amino acids have more than one folding shape, but
the maximum number would not be more than 27. Each folding shape for 5
amino acids, however, have different weight according the frequency of
appearance in PDB or the free energy for thermodynamic stability in
results of computational simulation. Thus, the folding shapes for each
set of 5 amino acids, which have higher frequency and lower free energy,
are considered with most probability in folding conformation, and are
assembled at the top rank in 5AAPFSC database.

According to protein sequence, the PFVM is constructed by extraction
local folding variations from 5AAPFSC database. The local folding
variations, which are represented by a set of PFSC letters for 5
successive amino acids, are extracted from 5AA-PFSC database and
displayed in vertical column. Along sequence from N-terminus to
C-terminus, the PFVM is formed. The procedure from a sequence to PFVM is
indicated by red arrows in Figure 1. A diagrammatic sketch in Figure 2
illustrated in detail how the local folding variations were assigned
according a protein sequence. The local folding shapes for each 5
consecutive amino acids along sequences are directly acquired from
database 5AAPFSC. Starting from the first set of 5 amino acid residues
at N-terminus, each step moves forward by one residue with sharing 4
amino acids from prior step until C-terminus. Therefore, the PFVM is
generated to reveal the local folding variations for entire protein, the
PFSC letters in each column show all possible local folding shapes for a
set of 5 amino acids which are extracted from 5AAPFSC database. With
PFVM, an astronomical number of PFSC strings can be constructed by
taking any one PFSC letter from each column. Also, the most possible
folding conformations can be predicted by the PFSC string which is
consisted of the PFSC letter on top of each columns in PFVM.
Furthermore, according PFSC string, the 3D structure of most possible
conformation can be constructed as the predicted result. Therefore, the
PFVM provides the significant information to study the protein fading
problem.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

Figure 2. Acquirement of Protein Folding Variation Matrix (PFVM). The
sequence is listed at the top horizontally. Then PFSC letters with color
in column, which represent local folding shapes of 5 consecutive amino
acids along sequences, are directly extracted from 5AAPFSC database.
Starting from N-terminus, each step moves forward by one residue sharing
4 amino acids next each other. The PFSC letters for each 5 successive
amino acids are vertically aligned down under the center of 5 amino
acids. The PFSC letters are marked by colors: red is for typical helix
fold; blue for typical beta fold; pink and light blue for folds with
partial helix or beta; black for irregular folds.

\textbf{All possible folding conformations.}

Astronomical number of folding conformations for a protein can be
constructed by the local folding variations in PFVM. To take one PFSC
letter from each column in the PFVM is able to form one PFSC string,
which is corresponding to one folding conformation. With fully using all
of local folding variations from each column, all possible folding
conformations will be constructed. Thus, the PFVM is one of the best
optimized approach to stores all folding information for entire protein.
The actual number of all possible folding conformations is able to be
figured out. Under PFSC scheme, an astronomical number of all possible
folding conformations for a protein is the product by multiplying the
numbers of PFSC letters of each columns in PFVM.

\(\text{Number\ of\ All\ Possible\ Conformations\ }=\ \prod_{i=1}^{n}{m(i)}=m\left(1\right)*m\left(2\right)*m\left(3\right)\ldots m(n)\)

Where \emph{n} is the number of columns in PFVM matrix and\emph{m}
(\emph{i} ) is the number of PFSC letters in column \emph{i} .

\textbf{RESULTS}

\textbf{PFVM Assembling Local Folding Variations.}

Based on protein sequence, the local folding variations of 5 successive
amino acids can be directly obtained and exhibited in the PFVM. All PFSC
letters for any 5 amino acids are extracted from the 5AAPFSC database,
and vertically aligned down under the center of corresponding 5 amino
acids. The order of PFSC letters in each vertical column is ranked
according to the probability of occurrence from higher to lower and the
free energy from lower to higher. Thus, the local folding variations for
an entire protein can be meaningfully displayed in PFVM. Two of human
proteins, small ubiquitin-related modifier 1(SUMO1)and cellular tumor
antigen p53 (P53), are taken as examples, and their PFVM will be
acquired according the sequences respectively.

\emph{PFVM for SUMO1 protein}

The protein of small ubiquitin-related modifier 1 (SUMO1)has highly
flexible on the regions of N- and C-terminuses, which involves a variety
of cellular processes, such as nuclear transport, transcriptional
regulation, apoptosis and protein stability. The sequence of human
protein SUMO1\_HUMAN with 101 of amino acids is available from UniProt
database. The PFVM was obtained within a few of seconds on a personal
computer (4 x {[}Intel(R) Core(TM) i53337 CPU @1.80 GHZ{]}, Windows
64bit operating system) after input of the sequence of SUMO1\_HUMAN. The
PFVM for SUMO1\_HUMAN protein is shown in Table 1. It is obvious that
each set of 5 amino acids has different folds. For example, starting
from N-terminate, the first set of 5 amino acids ``MSDQE'' has 11 PFSC
letters representing different folding shape. With moving forward by one
amino acid, the second set of 5 amino acids ``SDQEA'' has 13 PFSC
folding shapes; the third set of 5 amino acids ``DQEAK'' has 11 of
different folding shapes; fourth set of 5 amino acids ``QEAKP'' has 9
different folding shapes; fifth set of 5 amino acids ``EAKPS'' has 8
different folding shapes and so on. Except the numbers different, the
patterns of folding shapes for each set of 5 amino acids are different.
With PFSC scheme, the distribution of the different numbers of folding
shapes as well as the different types of folding shapes for entire
protein of SUMO1\_HUMAN can be simultaneously observed. Also, the
pattern of folding variations of SUMO1\_HUMAN in PFVM is overall agree
with the disorder determined by PDB which is displayed on the bottom of
Table 1, i.e. both N-terminus and C-terminus are more disorder than
center region. Thus, the PFVM actually demonstrates the folding
variations, and the local folding variations of protein SUMO1 relying on
the order of amino acids in its sequence.

The PFVM in Table 1 also revealed the features of folding conformation
for entire protein of SUMO1\_HUMAN. First, the PFVM is able to expose
the flexibility or rigidity for protein folding conformations. In the
middle portion of the sequence, the fragment (45-54), has a favor
conformation as ``AAAADAAAAA'' with PFSC letters in first row which is
almost typical alpha-helical conformation, and has the favor
conformation as ``DDDDADDDDD'' in second row which is alike
alpha-helical conformation. Also, the fragment in middle portion has
fewer options changing folding variations than the fragments at both
N-terminus and C-terminus. The revealed folding pattern in PFVM for
SUMO1\_HUMAN generally agrees with the knowledge from given protein 3D
structural data in PDB. Seven of 3D structures for SUMO1\_HUMAN protein
are displayed in Table 2, which listed their 3D images with PDB ID,
measurement methods, resolution, solvent and ligand, interacted protein,
PH and temperature etc. Thus, these 3D structures may have different
folding conformations. With comparison, the superimposition of 17
folding conformations from seven PDB 3D structures is displayed in
Figure 3, where the fragments in middle region share common folding
pattern while the fragments at both N-terminus and C-terminus are
diverse. Second, the PFVM is able to provide the information to
construct all possible conformations for a protein. Each conformation of
SUMO1\_HUMAN can be constructed by taking one PFSC letter from each
column in PFVM, and is expressed by a PFSC string. Apparently, the
conformations with astronomical number can be generated by various
combinations of PFSC letters in PFVM, and its astronomical number can be
calculated by multiplying the number of PFSC letters in each column for
each 5 amino acids. For SUMO1\_HUMAN, the astronomical number of all
possible conformations is 1.587x10\textsuperscript{77}. Obviously, it is
impossible to display all 3D folding structures of such huge number.
Thus, the PFVM truly is an optimized mode to present the comprehensive
local folding variations for entire protein, and it can generate all
possible folding conformations for a protein. Furthermore, the most
possible conformations with stable states can be constructed from PFVM.
One of most probable conformations for SUMO1\_HUMAN protein is directly
consisted by PFSC letters on the first row in PFVM of Table 1.
Therefore, the PFVM provides significant folding information for
SUMO1\_HUMAN.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

Table 1. The Protein Folding Variation Matrix (PFVM) for Human small
ubiquitin-related modifier 1 (SUMO1\_HUMAN) protein. Top section: the
sequence of 101 of amino acids of SUMO1\_HUMAN with numeric ruler.
Section below sequence: the PFVM. The PFSC letters in each vertical
column represent the local folding variation for 5 successive amino
acids along sequence. The order of PFSC letters in each vertical column
is ranked from higher to lower according the frequency numbers of
folding shapes in PDB. The PFSC letters are marked by colors: red is for
typical helix fold; blue for typical beta fold; pink and light blue for
folds with partial helix or beta; black for irregular folds. Bottom
section showed the disorder determined by PDB.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

Table 2. Experimental and environmental parameters for seven protein
structures of (SUMO1\_HUMAN) protein. Seven 3D structures are obtained
from PDB which were determined by NMR measurement or crystal X-ray
diffraction. Each structure was measured under different conditions,
such as method, resolution, solvent and ligand, protein interacted, PH
and temperature etc. The SUMO1 protein is marked in the color yellow in
images of structure complexes. The model index, chain name and fragment
position in sequence are indicated.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}

\end{center}
\end{figure}

Figure 3. The superimposition of conformations for SUMO1\_HUMAN. Seven
structures in PDB provide 17 conformations. 1A5R has ten conformations,
1Y8R has two conformations and 3KYD, 1WYW, 2PE6, 3KYC and 2BF8 has one
conformation individually. The ribbons are colored by segments as the
followings, black for loop, red for helix, blue for strand and light
blue for turn.

\emph{PFVM for P53 protein}

The protein of Human Cellular tumor antigen p53(P53\_HUMAN)is another
sample to illustrate how the local folding variations in PFVM correlate
with the order of amino acid along sequence. The p53 involves many
mechanisms for anticancer functions and plays a role in apoptosis,
genomic stability, and inhibition of angiogenesis. The human~TP53~gene
encodes 9 of protein isoforms11Surget S, Khoury MP, Bourdon JC,
''Uncovering the role of p53 splice variants in human malignancy: a
clinical perspective''.~OncoTargets and Therapy.~7: 57--68, (2013)., and
the canonical sequence of isoform p53-alpha with 393 amino acids is
available in UniProt database. According the sequence, the PFVM of
P53\_HUMAN protein is obtained and shown in Table 3. Although the
protein of p53-alpha for human has 393 amino acids in sequence, the PFVM
of p53-alpha was acquired within one minute using a personal computer.
First, the PFVM directly displayed the fluctuation of local folding
variations along the sequence. For examples, some sets of 5 successive
amino acids have only one folding shape representing with one PFSC
letter in PFVM, such as fragments in sequence of ``DEAPR'' (61-65),
``EAPRM'' (62-66), ``APSWP'' (88-92), ``PSWPL'' (89-93), ``NKMFC''
(131-135) and ``KMFCQ'' (132-136) etc. Some sets of 5 successive amino
acids have more folding shapes representing with higher number of PFSC
letters in PFVM, such as ``PLSQE'' (13-17) has 14 folding shapes;
``SDGLA'' (185-189) has 13; ``SSGNL'' (260-264) 15 and ``SKKGQ''
(371-375) 16 etc. Second, the pattern of folding variations of
P53\_HUMAN in PFVM showed many regions agree with the disorder
determined by PDB which is displayed on the bottom of Table 3. For
examples, regions of 20-25, 124-136, 230-241 and 332-347 have less
variations and disorder. However, the PFVM may exposed the folding
variations more detail. Third, all possible conformations for the
p53-alpha protein can be constructed by various combinations with taking
one PFSC letter from each column in PFVM. Actually, with PFVM, an
astronomical number of folding conformations for the p53-alpha protein
can be constructed and represented by 2.008x10\textsuperscript{270} of
PFSC strings. It showed that the astronomical number of all possible
conformations for p53-alpha protein were not vague anymore because the
number can be explicitly calculated based on PFVM.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}

\end{center}
\end{figure}

(Continue)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}

\end{center}
\end{figure}

(Continue)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}

\end{center}
\end{figure}

(Continue)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image9/image9}

\end{center}
\end{figure}

Table 3. Protein Folding Variation Matrix (PFVM) for protein of Human
cellular tumor antigen p53 (P53\_HUMAN). Top section: the sequence of
393 of amino acids of P53\_HUMAN with numeric ruler. Section below
sequence: the PFVM. The PFSC letters in each vertical column represent
the local folding variation for 5 successive amino acids along sequence.
The order of PFSC letters in each vertical column is ranked from higher
to lower according the frequency numbers of folding shapes in PDB. The
PFSC letters are marked by colors: red is for typical helix fold; blue
for typical beta fold; pink and light blue for folds with partial helix
or beta; black for irregular folds. Bottom section showed the disorder
determined by PDB.

The PFVM has capability to discover the change of local folding
conformation caused by mutation, even if 3D structural data is lacked.
The P53 has an important role as a~tumor suppressor, but the mutations
severely compromise the tumor suppression. The mutations may happen in
various positions or make different substitution in same position of
sequence related to different diseases and biological functions. The
different substitution of amino acid residue 183 of P53, for example,
related to different biological functions. The mutation S183E abolishes
phosphorylation, but the mutation S183A inhibits its transcriptional
activity.\textsuperscript{17} The replacement of amino acid residue
triggered the changes of folding conformation. The changes of local
folding shapes around residue 183 of P53\_HUMAN are revealed by PFVM and
the results are exhibited in Table 4. The mutation on a residue
primarily affects around five columns of local folding variations in
PFVM which are highlighted with a yellow color. It is apparent that with
comparison of native state, the folding patterns are changed and the
numbers of folding variations are reduced respectively after mutating
S183E and S183A in P53 sequence. The change of conformation, which is
caused by a residue replacement in mutation, is hard to be observed by
experimental measurements, such as NMR, X-ray crystallography or
Cryo-electron microscopy, and it is hard to be compared by the results
from computational molecule dynamics simulation. However, the PFVM
provided a useful means to reveal the difference of folding conformation
between before and after mutation. Additionally, the folding
conformation changes in PFVM caused by mutation are further demonstrated
that the protein folding was associated with the order of amino acids in
sequence.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image10/image10}

\end{center}
\end{figure}

Table 4. Comparison of Protein Folding Variation Matrix (PFVM) for
mutation on residue 183 of P53\_HUMAN. Top section: the sequence
fragment (170-200) of P53\_HUMAN with numeric ruler. Bottom section: the
local folding variation in PFSC letters. The PFSC letters in each
vertical column represent the local folding variation for 5 successive
amino acids in sequence. The order of PFSC letters in each vertical
column is ranked from higher to lower according the frequency numbers of
folding shapes in PDB. The PFSC letters are marked by colors: red is for
typical helix fold; blue for typical beta fold; pink and light blue for
folds with partial helix or beta; black for irregular folds.

\textbf{PFVM Embracing Known Structures}

The PFVM can generate all possible folding confirmations for a protein
with an astronomical number, of course, it should embrace the protein
conformations of known 3D structures. For a protein, it usually may have
multiple structural data in PDB, but these 3D structures in some degrees
have difference in folding conformations because of different
measurement methods and environments. With PFVM, different conformations
in PFSC strings can be constructed by the PFSC letters from each column
in matrix, and the conformations for protein with known 3D structures
should be one of these PFSC strings.

\emph{SUMO1 known 3D structure conformations covered by PFVM}

The SUMO1\_HUMAN protein has more than 50 of 3D structures available in
PDB, and it is not surprised that these structures have both similarity
and difference in folding conformations. Here, seven of 3D structures of
SUMO1\_HUMAN (Table 2) as examples are taken to compare the folding
conformations. First, these 3D structures were obtained by different
measurement methods, under different environments and interaction with
different molecules. Second, many conformations can be generated from
these protein 3D structures. 1A5R has 10 of conformations which is NMR
experimental data in water solution.11Bayer, P., Arndt, A., Metzger, S.,
Mahajan, R., Melchior, F., Jaenicke, R., Becker, J. Structure
determination of the small ubiquitin-related modifier SUMO-1 {[}J{]}.
Journal of Molecular Biology, 280(2):275, (1998). The rest of protein
complex structures were measured by X-ray crystal diffraction, 1Y8R has
two chains and 3KYD, 1WYW, , 2PE6, 3KYC and 2BF8 has one chain
respectively22Olsen SK1, Capili AD, Lu X, Tan DS, Lima CD.,Active site
remodelling accompanies thioester bond formation in the SUMO E1, Nature.
463(7283):906-12, (2010).\textsuperscript{,}33Baba D, Maita N, Jee J G,
et al. Crystal structure of thymine DNA glycosylase conjugated to
SUMO-1{[}J{]}. Nature, 435(7044):979-82,
(2005).\textsuperscript{,}44Lois L M, Lima C D. Structures of the SUMO
E1 provide mechanistic insights into SUMO activation and E2 recruitment
to E1{[}J{]}. Embo Journal, 24(3):439-51,
(2005).\textsuperscript{,}55Capili A D, Lima C D. Structure and Analysis
of a Complex between SUMO and Ubc9 Illustrates Features of a Conserved
E2-Ubl Interaction{[}J{]}. Journal of Molecular Biology, 369(3):608-618,
(2007).\textsuperscript{,}66Pichler A, Knipscheer P, Oberhofer E, et al.
SUMO modification of the ubiquitin-conjugating enzyme E2-25K{[}J{]}.
Nature Structural \& Molecular Biology, 12(3):264-9, (2005).. Thus,
these seven SUMO1 protein structures contain total of 17 folding
conformations, and they are respectively converted into 17 of PFSC
strings according their coordinates of alpha C-atoms. The PFSC strings
for fragment (21-84) for 17 folding conformations of SUMO1\_HUMAN are
aligned and listed in the top section of Table 5.

It is easy to compare the similarity or dissimilarity of structure
conformations with alignment of PFSC alphabetical strings. We try to
find out how many of different types of local folding shapes in each
column exist for 17 of folding conformations for seven 3D structures. In
order to do so, all PFSC letters for conformation of 1A5R-01 (model 1)
are firstly counted and marked by yellow background color. Then, for the
rest of 16 PFSC strings, any PFSC letter in same column not same as
1A5R-01 or other will be highlighted by yellow. Thus, the PFSC letters
with yellow color on each column will reveal what different types of
folding shapes exist for each of 5 successive amino acid residues. For
the 17 known conformations, the fragment (21-84) in SUMO1 has 64
columns, i.e. 64 sets of 5 successive amino acid residues in structure.
There are 9 columns which have the identical local folding shapes as
1A5R-01, and the remainder 55 columns have at least one different local
folding shape from 1A5R-01. Therefore, it is not too hard to detect the
similarity or dissimilarity for 17 conformations with alignment of PFSC
strings. First, all 17 conformations have similar secondary structures
distributed along sequence which is observed by the PFSC letter colors.
Second, none of 16 conformations are matching 1A5R-01. Third, it is
apparent that each structure has unique folding conformations and can be
distinguished from each other. Specifically all six structures from
X-ray crystallography have larger alterations of local folds than
1A5R-01 conformations from NMR measurement. In summary, despite overall
similarity of secondary structure, the known 3D structures of SUMO1
protein have different folding conformations which are well revealed by
alignment of PFSC strings.

The PFVM of SUMO1\_HUMAN protein contains the comprehensive folding
information which embraces all 17 folding conformations of known
structures. The PFVM for SUMO1\_HUMAN protein is displayed on the bottom
section of Table 5. The PFSC letters on each column represent the
possible folds for each 5 successive amino acids. In order to show the
PFVM covering the folds in the known structures, any PFSC letters in
PFVM, which already have appeared on same column for known structures,
are marked by yellow. It is apparently that all folding letters with
yellow in the known structures are enclosed by the PFVM. For example,
the PFVM in column 38 has 3 types of folds (W, B and L) covering the
folds of ``W'' and ``L'' in given structures; the PFVM in column 39 has
4 types of folds (C, S, W and R) covering the folds of ``C'', ``S'' and
``R'' in given structures; the PFVM in column 40 has 6 types of folds
(Y, S, V, Z, B and C) covering the folds of ``V'', ``B'' and ``Y'' in
the given structures and etc. Thus, the PFVM has complete local folding
variations, which is able to cover the folding changes in given 3D
structures of SUMO1 protein. Furthermore, the most possible
conformations are able to be predicted by PFVM. The PFSC string on first
row of PFVM in Table 5, which is comprised of the local folding shapes
with the most tendencies, directly present one of the most possible
conformations for SUMO1 protein. With yellow colors of PFSC letters, it
is apparent that the predicted conformation is overall aligned well with
all of 17 conformations from known 3D structural. Also, the most
possible conformation has 60 among 64 PFSC letters with the marked
yellow color, which indicated the most possible conformations is matched
with the known structures. In conclusion, the PFSC string on first row
in PFVM provides one of well predicted conformations. Also,
comprehensive local folding variations in PFVM are able to cover the
various conformations of SUMO1 protein from known 3D structures.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image11/image11}

\end{center}
\end{figure}

Table 5. The PFVM of SUMO1\_HUMAN embracing folding conformations of
protein with known 3D structures. Top section: the sequence (21-84) of
SUMO1\_HUMAN protein with numeric ruler. Middle section: the 17 PSCF
strings for conformations of 7 known 3D structures of SUMO1\_HUMAN.
Seven of structures are available in PDB: 1A5R is NMR data with 10
models; 3KYD, 1WYW, 1Y8R, 2PE6, 3KYC and 2BF8 are crystal X-ray
diffraction data with one chain respectively and 1Y8R with two chains.
The all PFSC letters of 1A5R-01 is first marked by yellow color. Other
PFSC letters in same column for these given structures are marked by
yellow color if they differ from 1A5R-01. Bottom section: the local
folding variation with PFSC letters for PFVM of SUMO1\_HUMAN protein.
The PFVM of SUMO1\_HUMAN protein. The PFSC letters in each column in
PFVM are highlighted by yellow color if the corresponding local folding
shapes for 5 successive amino acids in given 3D structures are yellow.
The PFSC letters are marked by colors: red is for typical helix fold;
blue for typical beta fold; pink and light blue for folds with partial
helix or beta; black for irregular folds.

\emph{P53 known 3D structure conformations covered by PFVM}

Many P53\_HUMAN structures are available in PDB from X-ray
crystallization measurements. Twelve of protein 3D structures of P53
(4AGP, 2YBG, 2XWR, 2X0U, 2J1X, 2J1Y, 2FEJ, 2BIN, 3D05, 3D06, 3ZME and
5LAP) are displayed in Figure 4. The PSCF strings as folding
conformations are obtained directly according 3D structures of regions
(101-200) in sequence of 12 given P53 structures, and are aligned and
displayed on the top section of Table 6. Also, the PFVM as local folding
variations is obtained according merely the sequence (101-200) of P53
and 5AAPFSC database, and is displayed on the bottom section.

The PFVM is able to cover all conformations of given 3D structures of
P53 protein. In order to verify what types of local folding shapes occur
in these given P53 structures, all PFSC letters in 2YBG are first
highlighted by yellow as starting point in Table 6, and then the PFSC
letters for other 3D structures are checked down each column against
2YBG. If the PFSC letters are not covered by 2YBG, the letters will be
highlighted by yellow. Thus, all local folding variations existing in
these given structures are shown by yellow. Furthermore, to find out
whether the folding conformations of 12 given structures are enclosed by
PFVM, the PFSC letters in each column in PFVM are marked by yellow if
the PFSC letter in same column appears in given 12 structures.
Therefore, it is easy to observe the local folding conformations of 12
given structures are indeed enclosed by PFVM. For examples, the PFVM in
column 104 provides 8 types of folding shapes (B, V, U, E, Y, C, A and
W) covering the folds of ``B'' and ``V'' in the given structures; the
PFVM in column 105 provides 4 types of folding shapes (W, A, Y and P)
covering the folds of ``W'' and ``A'' in the given structures; the PFVM
in column 120 provides 11 types of folding shapes (Y, A, V, P, Z, E, J,
\$, B, D and R) covering the folds of ``Y'' and ``Z'' in the given
structures; the PFVM in column 139 provides 7 types of folding shapes
(W, P, A, S, B, V and J) covering the folds of ``P'' and ``W'' in the
given structures; the PFVM in column 199 provides 6 types of folding
shapes (A, J, Y, C, P and S) covering the folds of ``A'', ``Y'' and
``J'' in the given structures and etc. In conclusion, the PFVM is able
to cover the local folding shapes of 12 given 3D structures for P53
protein, and the PFVM reserves far more local folding variations for P53
protein in other environments.

The PFVM provides the information to predict the possible fording
conformation for protein. Despite a massive number of folding
conformations for P53 protein can be generated with the combination of
various PFSC letters in PFVM, the PFSC string at the first row in PFVM,
which is named as PFVM-01, represents one of the most possible
conformations for P53 protein. The PFVM-01 can be compared with the
conformations of 12 given 3D structures in Table 6. Overall, with
observation in color, the secondary structure fragments of PFVM-01
conformation are well aligned with all 12 conformations of given 3D
structures of P53 protein. Also, 99 among 100 PFSC letters in the
PFVM-01 conformation appear in the 12 given 3D conformations of P53
protein. Therefore, the PFSC string in the first row of PFVM provides
one of best conformations for P53 protein prediction.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image12/image12}

\end{center}
\end{figure}

Figure 4. 3D structure images of folding conformations of 12 structures
for P53\_HUMAN protein displayed with solid ribbon style. The structure
data was obtained from PDB and the PDB ID is listed below the image.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image13/image13}

\end{center}
\end{figure}

Table 6. The PFVM of P53\_HUMAN embracing folding conformations of known
3D structures. Top section: the sequence (101-200) of P53\_HUMAN protein
with numeric ruler. Middle section: the PSCF strings for 12 known 3D
structures of P53\_HUMAN which are available in PDB: 2YBG, 4AGP, 2XWR,
2X0U, 2J1X, 2J1Y, 2FEJ, 2BIM, 3D05, 3D06, 3ZME and 5LAP. All PFSC
letters for 2YBG is first marked by yellow color. Other PFSC letters in
same column for these given 3D structures are marked by yellow color if
they differ from 2YBG. Some pieces of missed fragments without atomic
coordinate data in structure database in PDB are indicated by dots.
Bottom section: the PFVM of P53\_HUMAN protein. The PFSC letters in each
column in PFVM are highlighted by yellow color if the corresponding
local folding shapes for 5 successive amino acids in given 3D structures
are yellow. The PFSC letters are marked by colors: red is for typical
helix fold; blue for typical beta fold; pink and light blue for folds
with partial helix or beta; black for irregular folds.

\textbf{Astronomical Number of Conformations}

The number of total conformations for a protein can be figured out
according the aggregation of local folding variations in the PFVM.
Generally known, total number of possible folding conformations for
protein is large, but the so called astronomical number of conformations
for any protein is blurred. However, with PFVM, the number of total
conformations for a protein can be explicitly calculated based on the
PFVM because total number of conformations for a protein is the product
of numbers of PFSC letters at each column in PFVM. The total numbers of
all possible conformations for ten proteins as samples are listed in
Table 7. These samples represent a wider variety of proteins and the
lengths of sequences with a range from 101 to 1,382 of amino acids. The
results show that the total numbers of possible conformations for these
proteins are large which have a scope from 10\textsuperscript{77} to
10\textsuperscript{784}. Two proteins among these samples, PDCD1\_MOUSE
and PDCD1\_HUMAN, belong to same gene code PDCD1 with equal length of
sequence, but they are respectively for human and mouse species. Despite
both proteins having the equal length of sequence of 288 amino acids,
the alignment revealed 172 of amino acids identical (about 60\%) between
two protein sequences. According PFVM of both proteins, total numbers of
conformations respectively are 3.028x10\textsuperscript{151} and
2.827x10\textsuperscript{147} because of the protein differentiation in
sequence for different organisms. The results demonstrated that the
total number of conformations of protein related to the length of
sequence as well as the composition of amino acids in sequence. In
summary, based on protein sequence, the PFVM does not only expose the
comprehensive local folding variations and predicted the most possible
conformations, but also provided the actual number of total
conformations for a protein.\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
\begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{UniProt Entry}\strut
\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{Protein names}\strut
\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{Biology Function \& Disease Related}\strut
\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{AA Length}\strut
\end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright\strut
\textbf{Total Number}\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
SUMO1\_HUMAN\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Small ubiquitin-related modifier 1\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Cellular processes, such as nuclear transport, transcriptional
regulation, apoptosis, and protein stability\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
101\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
1.587x10\textsuperscript{77}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
INS\_HUMAN\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Insulin\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Insulin decreases blood glucose concentration;Diabetes mellitus\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
110\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
4.940x10\textsuperscript{96}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
SYUA\_HUMAN\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Alpha-synuclein\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Regulation of dopamine release and transport; Induces fibrillization of
microtubule-associated protein tau; Parkinson disease\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
140\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
2.385x10\textsuperscript{126}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
TNFA\_HUMAN\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Tumor necrosis factor\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Induce cell death of certain tumor cell lines; Impairs regulatory
T-cells function; Immune system diseases; Rheumatoid arthritis\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
233\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
3.028x10\textsuperscript{151}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
PDCD1\_MOUSE\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Programmed cell death protein 1\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Inhibitory cell surface receptor involved in the regulation of T-cell
function during immunity and tolerance\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
288\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
2.827x10\textsuperscript{147}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
PDCD1\_HUMAN\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Programmed cell death protein 1\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Inhibitory cell surface receptor involved in the regulation of T-cell
function during immunity and tolerance\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
288\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
1.762x10\textsuperscript{155}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
P53\_HUMAN\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Cellular tumor antigen p53\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
A tumor suppressor in many tumor types; induces growth arrest or
apoptos; Cancer.\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
393\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
2.008x10\textsuperscript{270}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
A4\_HUMAN\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Amyloid-beta A4 protein\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
neurons relevant to neurite growth, neuronal adhesion and axonogenesis;
Alzheimer disease\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
770\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
1.875x10\textsuperscript{460}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
EGFR\_HUMAN\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Epidermal growth factor receptor\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Receptor tyrosine kinase binding ligands of the EGF family and
activating several signaling cascades to convert extracellular cues into
appropriate cellular responses; Non-small cell lung cancer\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
1,210\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
5.693x10\textsuperscript{743}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
INSR\_HUMAN\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Insulin receptor\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
Receptor tyrosine kinase which mediates the pleiotropic actions of
insulin; Diabetes mellitus\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
1,382\strut
\end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright\strut
2.232x10\textsuperscript{784}\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}

Table 7. Total number of possible folding conformations for ten proteins
respectively. The protein names with other information were obtained
from UniProt database. Total number: the number of folding conformations
was obtained according PFVM. The PFVM for these ten proteins are listed
in the supplemental file.

\textbf{Prediction of Most Possible Conformation and 3D Structure}

The most possible conformation and 3D structure for protein can be
predicted from its PFVM. With a protein sequence, the local folding
variations are collected in PFVM. For examples, the PFVM of SUMO1\_HUMAN
is at Table 1; the PFVM of P53\_HUMAN at Table 3 and the PFVM of
K4GSD6\_9SAUR, C4IXC1\_9TELE, A0A851ZE52\_9AVES and EP3B\_HUMAN in
supplementary document. The alphabetic PFSC string on top of PFVM, which
is named as PFVM-01, represents the most possible folding conformation.
Their PFVM-01 are listed on Table 8, which are the most possible folding
conformations for proteins. In PFVM-01, each of PFSC letter represents
the folding shape of 5 amino acids in sequence, two PFSC letters next
each other share 4 amino acids, and then each PFVM-01 is a PFSC string
for folding conformation from N-terminus to C-terminus. As the PFSC
letters in PFVM-01 are on top folding shapes in PFVM, the PFVM-01
represents the most possible conformation for a protein.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image14/image14}

\end{center}
\end{figure}

Table 8. The most possible folding conformations of PFVM-01. Left column
listed the UniprotKB and protein name; right column listed the PFVM-01
which is the PFSC string on top of PFVM. The PFVM of SUMO1\_HUMAN is at
Table 1; the PFVM of P53\_HUMAN at Table 3 and the PFVM of
K4GSD6\_9SAUR, C4IXC1\_9TELE, A0A851ZE52\_9AVES and EP3B\_HUMAN in
supplementary document.

With PFVM-01, the protein 3D structure can be constructed which
represents the predicted structure. The PFVM-01 is a PFSC string, which
represented the folding conformation for a protein. First, with
searching for similar conformations, an initiative 3D structure can be
constructed according an entire PFVM-01. Using high throughput screening
PDB, the similar structural conformations can be obtained according
folding similarity score with comparison between the PFVM-01 and all
PFSC strings in database. Thus, the initiative 3D structure can be
constructed according the 3D structure with highest folding similarity
score. Second, the parts of PFVM-01, particularly tertiary fragments,
can be further searched to improve 3D structure. Third, the side chains
are added for each residue. Finally, the constructed 3D structure is
optimized by computed for free energy minimization. With PFVM-01, six of
3D structures are predicted and displayed in Figure 5. First row
displays the predicted 3D Structure from PFVM-01 of SUMO1\_HUMAN; second
row displays the predicted 3D Structure from PFVM-01of P53\_HUMAN. The
SUMO1\_HUMAN and P53\_HUMAN have the known 3D structures, so the
comparison between known 3D structure and the predicted structure (in
brown color) are shown on left side. Although proteins of K4GSD6\_9SAUR,
C4IXC1\_9TELE, A0A851ZE52\_9AVES and EP3B\_HUMAN do not have 3D
structures available in PDB, their most possible 3D structures can be
predicted by PFVM-01 are displayed respectively in Figure 5.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image15/image15}

\end{center}
\end{figure}

Figure 5. The predicted protein 3D structures from PFVM-01. First row
displays the predicted 3D Structure from SUMO1\_HUMAN PFVM-01; second
row displays the predicted 3D Structure from P53\_HUMAN PFVM-01; left
side is comparison between known 3D structure and the predicted
structure (brown color). The predicted 3D structures for K4GSD6\_9SAUR,
C4IXC1\_9TELE, A0A851ZE52\_9AVES and EP3B\_HUMAN are displayed
respectively.

\textbf{DISCUSSION}

\textbf{Computation and database for protein folding.}

Many of computational methodologies and database for protein folding
have been developed,11Compiani M, Capriotti E, ''Computational and
theoretical methods for protein folding''. Biochemistry. 52 (48):
8601--24, (2013).. and the efforts may be divided into two aspects, one
aspect is to predict the protein structure with thermodynamic stability
and another aspect is to investigate the protein conformations with
variability.

In first aspect, the prediction of protein structure from a sequence is
pursuing to obtain a native folding conformation with thermodynamic
stability, and the stable structure is mainly controlled by hydrophobic
interactions, hydrogen bonds, van der Waals forces, and conformational
entropy. In general, the methods for prediction of protein structure
fall into two main categories: template-free modeling and template-based
modeling.22Guo JT, Ellrott K, Xu Y. A historical perspective of
template-based protein structure prediction. Methods Mol Biol;
413:3--42, (2008).\textsuperscript{,}33Dorn M, E Silva MB, Buriol LS,
Lamb LC. Three-dimensional protein structure prediction: methods and
computational strategies. Comput Biol Chem; 53PB:251--76,
(2014).\textsuperscript{,}44Brylinski M. Is the growth rate of Protein
Data Bank sufficient to solve the protein structure prediction problem
using template-based modeling? : Bio-Algorithms and Med-Systems{[}J{]}.
Bio-Algorithms and Med-Systems, 11(1):1-7, (2015). The template-free
methods, i.e., \emph{ab initio} or \emph{de novo}approaches, are based
on the energy functions which carry out through the molecular dynamics
(MD) simulation calculations under various force fields for atoms
interaction or experiential parameters for group atoms
interaction.55Honig B. Protein folding: from the levinthal paradox to
structure prediction. J Mol Biol; 293:283--93,
(1999).\textsuperscript{,}66Onuchic JN, Wolynes PG. Theory of protein
folding. Curr Opin Struct Biol; 14:70--5,
(2004).\textsuperscript{,}77Zhang J, Li W, Wang J, Qin M, Wu L, Yan Z,
et al. Protein folding simulations: from coarse-grained model to
all-atom model. IUBMB Life; 61:627--43, (2009). The protein with stable
conformation is finally obtained by iterative convergence to lower
thermodynamic free energy under defined force fields, such as
AMBER,88Yang, L., Tan, C. H., Hsieh, M. J., Wang, J., Duan, Y., Cieplak,
P., Caldwell, J., Kollman, P. A., and Luo, R. New-generation amber
united-atom force field. J. Phys. Chem. B 110, 13166-13176, (2006).
CHARMM99Brooks, B. etc, CHARMM: The biomolecular simulation program. J.
Comput. Chem. 30, 1545-1614, (2009). and GROMOS1010Riniker, S., Christ,
C. D., Hansen, H. S., Hunenberger, P. H., Oostenbrink, C., Steiner, D.,
and van Gunsteren, W. F. Calculation of relative free energies for
ligand-protein binding, solvation, and conformational transitions using
the GROMOS software. J. Phys. Chem. B 115, 13570-13577, (2011). force
fields. The software from Chemistry at Harvard Macromolecular
Mechanics~(CHARMM)~\textsuperscript{33},1111Brooks BR, Bruccoleri RE,
Olafson BD, States DJ, Swaminathan S, Karplus M, ''CHARMM: A program for
macromolecular energy, minimization, and dynamics calculations''.~J Comp
Chem.~4~(2): 187--217, (1983).~ is one of the most mature algorithm for
molecular dynamics, which minimizes the free energy of a protein
structure while collecting the molecular dynamics trajectory of
united-atom all-atom, dihedral potential corrected variants and
polarization. The Rosetta software1212Leaver-Fay, A., Tyka, M., Lewis,
S. M., Lange, O. F., Thompson, J., Jacak, R., Kaufman, K., Renfrew, P.
D., Smith, C. A., Sheffler, W., Davis, I. W., Cooper, S., Treuille, A.,
Mandell, D. J., Richter, F., Ban, Y. E., Fleishman, S. J., Corn, J. E.,
Kim, D. E., Lyskov, S., Berrondo, M., Mentzer, S., Popovic, Z.,
Havranek, J. J., Karanicolas, J., Das, R., Meiler, J., Kortemme, T.,
Gray, J. J., Kuhlman, B., Baker, D., and Bradley, P., ROSETTA3: An
object-oriented software suite for the simulation and design of
macromolecules. Methods Enzymol. 487, 545-574, (2011). developed by
the~Berkeley Open Infrastructure for Network Computing~Platform is one
of \emph{de novo} tools to predict protein structure, which is assembled
by Monte Carlo simulated annealing procedure relying on a library of
residue fragments.1313Kroese, D. P.; Brereton, T.; Taimre, T.; Botev, Z.
I., ''Why the Monte Carlo method is so important today''.~WIREs Comput
Stat.~6: 386--392, (2014). In practice, the protein structure prediction
is efficient for calculating smaller proteins, and requires vast
computational resources for larger proteins. The template-based methods,
such as homology modeling or comparative modeling, align sequences
according to similarity of multiple templates from PDB, and then process
energy optimization to predict protein 3D structure. With sequence
homologous, it assumes that similar sequences have similar folding
conformations. Depending on the homology modeling, I-TASSER1414Roy, A.,
Kucukural, A., and Zhang, Y., I-TASSER: A unified platform for automated
protein structure and function prediction. Nat. Protoc. 5, 725-738,
(2010)., Robetta1515Kim, D. E., Chivian, D., and Baker, D., Protein
structure prediction and analysis using the Robetta server. Nucleic
Acids Res. 32, W526-W531, (2004). and MODELER1616Eswar, N., Webb, B.,
Marti-Renom, M. A., Madhusudhan, M. S., Eramian, D., Shen, M. Y.,
Pieper, U., and Sali, A., Comparative protein structure modeling using
MODELLER. Current Protocols in Protein Science, Chapter 2, Unit 2.9,
(2007), Wiley, New York.\textsuperscript{,}1717Liu, T., Tang, G. W., and
Capriotti, E., Comparative Modeling: The state of the art and protein
drug target structure prediction. Comb. Chem. High Throughput Screening
14, 532-537, (2011). software build protein for unknown 3D structure. If
there is not a distinguishably similar sequence matched in PDB database,
the template-free approaches will provide the supplement for
thermodynamics calculations. Recently, with a deep learning in
artificial intelligence (AI), AlphaFold approach was particularly
successful at predicting the most accurate structure and with
demonstration in CASP13 and CASP14.1818DeepMind's protein-folding AI has
solved a 50-year-old grand challenge of biology.~MIT Technology Review.
Retrieved~(2020).,1919Sample, Ian (2 December 2018).~''Google's DeepMind
predicts 3D shapes of proteins''.~The Guardian. Retrieved~30
November~(2020).,2020~''DeepMind's protein-folding AI has solved a
50-year-old grand challenge of biology''\emph{.~MIT Technology Review} .
(2020)\emph{.} AlphaFold first handled the protein structure as a
spatial graph with the residues as nodes and the connection of
residues~as edges. Then, it trained the system on all available protein
3D structures from PDB~together with~the databases~containing protein
sequences of unknown structure. For physical interactions within
proteins, it created an attention-based neural network system, and
trained residue-to-residue and atom-atom using an internal confidence
measure. The protein structure was refined by evolutionarily related
multiple sequence alignment (MSA) and a representation of amino acid
residue pairs. With iterating process, AlphaFold predicted the
underlying physical structure of the protein and is able to determine
highly-accurate structures.

In second aspect, the objective of protein folding is to investigate
variations of conformations because the proteins in essence are
non-static structures, but rather conformational ensembles with multiple
states. With general knowledge, the protein adjusts the folding
conformations under different environments or interaction of ligand or
protein. Also, intrinsically disordered proteins and regions (IDPs/IDR)
are widely distributed in natural proteins, which are associated with
many biological processes and diseases.2121Chen J , Guo M , Wang X , et
al., A comprehensive review and comparison of different computational
methods for protein remote homology detection{[}J{]}. Briefings in
Bioinformatics(2):2. 1--17, (2017). The IDPs/IDR for protein 3D
structures can be identified by many experimental techniques.2222Robin
van der Lee , etc. Classification of intrinsically disordered regions
and proteins.{[}J{]}. Chemical Reviews, 114(13):6589, (2014).
DisProt,2323Piovesan D, Tabaro F, Micetic I, et al. DisProt 7.0: a major
update of the database of disordered proteins. Nucleic Acids Res,
45:D1123--4, (2017). IDEAL2424Fukuchi S, Sakamoto S, Nobe Y, et al.
IDEAL: intrinsically disordered proteins with extensive annotations and
literature. Nucleic Acids Res; 40:D507--11, (2012). and MobiD2525Potenza
E, Di Domenico T, Walsh I, et al. MobiDB 2.0: an improved database of
intrinsically disordered and mobile proteins. Nucleic Acids Res
2015;43:D315--20. are useful databases for IDP/IDR, and PDB also
provides the illustration. Moreover, under physiological conditions, a
native protein essentially is able to undergo a reversible transition
between disorder and order folding conformations. In 1973, Anfinsen's
Nobel prize-winning experiments2626Anfinsen CB, Principles that govern
the folding of protein chains. Science 181: 223--230. showed that the
protein ribonuclease can be reversibly denatured and re-natured in a
test tube, and then over thousands of other proteins have been
demonstrated with folding reversibility with condition changes. The
protein has folding reversibility because of small energy barriers (5 to
15 kcal/mol) between the folded and unfolded populations.2727Kumar MD,
Bava KA, Gromiha MM, Prabakaran P, Kitajima K, Uedaira H, Sarai A (2006)
Nucleic Acids Res 34:D204--D206, (1973). Different computational
approach have been developed focusing on the variability of protein
folding. In the late 1970s, Karplus and Weaver developed the
diffusion-collision (DC) model,2828Karplus, M., and Weaver, D. L.,
Protein-folding dynamics. Nature 260, 404-406, (1976).,2929Karplus, M.,
and Weaver, D. L., Diffusion-collision model for protein folding.
Biopolymers 18, 1421-1437, (1979).,3030Islam, S. A., Karplus, M., and
Weaver, D. L., Application of the diffusion-collision model to the
folding of three-helix bundle proteins. J. Mol. Biol. 318, 199-215,
(2002).,3131Myers, J. K., and Oas, T. G., Preorganized secondary
structure as an important determinant of fast protein folding. Nat.
Struct. Biol. 8, 552-558, (2001). that explored the long-term protein
evolution and allowed the large amplitude changes in the folding
dynamics. Later it was modified into the foldon diffusion-collision
(FDC)3232Fuxreiter, M., Simon, I., Friedrich, P., and Tompa, P.,
Preformed structural elements feature in partner recognition by
intrinsically unstructured proteins. J. Mol. Biol. 338, 1015-1026,
(2004).,3333Compiani, M., Capriotti, E., and Casadio, R., Dynamics of
the minimally frustrated helices determine the hierarchical folding of
small helical proteins. Phys. Rev. E: Stat., Nonlinear, Soft Matter
Phys. 69, 051905, (2004).,3434Stizza, A., Capriotti, E., and Compiani,
M., A minimal model of three-state folding dynamics of helical proteins.
J. Phys. Chem. B 109, 4215-4226, (2005). which provided a more refined
description of folding transforms, including predicting the secondary
native structure and specifying stability of the foldons themselves. In
1977, the hydrophobic collapse (HC) mechanism3535Dill, K. A., Theory for
the folding and stability of globular proteins. Biochemistry 24,
1501-1509, (1985).,3636Haran, G., How, when and why proteins collapse:
The relation to folding. Curr. Opin. Struct. Biol. 22, 14-20, (2012).
was developed to predict that the hydrophobic forces and backbone forces
result in chain collapse prior to the formation of elements of secondary
structure. Of course, except hydrophobic, the hydrogen bonds and van der
Waals forces are also steering the unfolded protein toward a collapsed
configuration.3737Barbosa, M. A., Garcia, L. G., and Pereira de Araujo,
A. F., Entropy reduction effect imposed by hydrogen bond formation on
protein folding cooperativity: Evidence from a hydrophobic minimalist
model. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 72, 051903,
(2005). In 2000, Folding@Home project was developed at Stanford
University to compute the protein folding with widely adopting the
contribution of computing resource. As a huge number of folding
conformations, the molecular dynamics (MD) simulations is a
time-demanding process which rely on parallel supercomputing
architectures or using personal computing clusters.3838Zagrovic, B.,
Snow, C. D., Shirts, M. R., and Pande, V. S., Simulation of folding of a
small \selectlanguage{greek}α-\selectlanguage{english}helical protein in atomistic detail using worldwide-distributed
computing. J. Mol. Biol. 323, 927-937, (2002).,3939Adcock, S. A., and
McCammon, J. A., Molecular dynamics: survey of methods for simulating
the activity of proteins.Chem. Rev. 106, 1589-1615, (2006).,4040Rizzuti,
B., and Daggett, V., Using simulations to provide the framework for
experimental protein folding studies. Arch. Biochem. Biophys. 531,
128-135, (2013).,4141Daggett, V., Protein folding-simulation. Chem. Rev.
106, 1898-1916, (2006). Anyway, the computational approaches for all
possible conformations to thoroughly resolve the protein folding problem
is now far less successful than was thought in the early days, and it is
still one of challenging subjects in the field of protein physical
science. Recently,~Google's~DeepMind applied the artificial intelligence
(AI) and successfully developed Alphafold approach which can regularly
predict protein structures with atomic accuracy competitive with
experimental structures. It trained a neural network to accurately
predict the distances between pairs of residues in a protein, and a
protein was optimized by a simple gradient descent algorithm to realize
structures. As the achievement of Alphafold, more scientific resource
and attention are focusing on the resolution of protein folding problem.

The protein folding information can be extracted from protein structure
databases. The PDB is the most inclusive repository of protein 3D
structures. So far, nearly 190,000 protein 3D structures have been
available in PDB, where approximately 90\% are obtained by X-ray
crystallography and the remain by NMR, CryoTEM and other techniques. The
X-ray crystallography may determine accurate atomic coordination for 3D
structure, but it only represents a specific static protein folding
state. The NMR~and CryoTEM display the protein flexibility that
structural oscillation is limited around an equilibrium state under
certain conditions. The Structure Classification of Proteins
(SCOP)4242http://scop.mrc-lmb.cam.ac.uk/scop database classifies the
protein structural domains into the hierarchy in terms of Species,
Protein, Family, Superfamily, Fold and Class. It defines 1,232 folds,
2,026 superfamilies and 4,919 families. If two protein domains have
similar secondary structures with the similar topological connections,
they belong to the same fold. The Class, Architecture, Topological fold
and Homologous superfamily (CATH) 4343http://www.cathdb.info classifies
95 million of protein domains into 1,391 topological folds and 6,119
superfamilies. If two proteins have similar topological fold and
sequence in conjunction with similar functions, they are assumed to be
associated with the same category in CATH. The
ProTherm4444http://www.abren.net/protherm/ database is a source for
understanding the protein folding stability with the thermodynamic
parameters for 25,830 structures, which includes numerical data changes
in Gibbs free energy, enthalpy, heat capacity and transition
temperature~etc. Nevertheless, the crucial question is whether the
protein database can be directly utilized for the investigation of
protein folding. The first question is if current protein structural
data and future coming data are sufficient for fold recognition, and the
answer is negative.\textsuperscript{28} The second question is whether
the defined topological folding patterns (about 1200 types of folds in
SCOP and near 1,400 in CATH database) are enough to correlate the
protein folding with the regulation of amino acid in sequence, and the
answer is insufficient. However, a number of structural data from
experimental and computational approaches should assist to understand
the protein folding in some degree. As a whole, the longer fragments
were hard thorough to investigate the folding patterns because of the
larger the folding prototype involving less universal folding pattern.
Therefore, to define a universal small folden as a prototype, such as
the backbone of 5 amino acid residues, may overcome these obstacles to
probe the folding patterns in protein structure database.

Here, the protein structure fingerprint approach demonstrated a useful
means to describe complete protein folding conformations and to
construct explicit database for protein folding. In mathematical space,
the backbone of 5 points connection is adopted as a universal folden and
the complete folding space is described by 27 PFSC alphabetic letters.
In biological space, the possible folds of 5 of amino acid residues are
limited by constrains, and then different combinations of 5 of amino
acid residues have different folding number and patterns. Thus, a
database (5AAPFSC) was created to collect all folding shapes for all
combinations of 5 of amino acid residues. For protein, one PFSC string
represents a complete folding description, and one PFVM matrix
represents comprehensive folding variation. Based on PFVM, not only does
all possible folding conformations in astronomical number are obtained,
but the most possible conformations are also obtained. Therefore, the
protein structure fingerprint approach covers two aspects, it can
predict stable folding conformation as well discover variations of
folding conformation with massive number. Furthermore, the digital
alphabetic PFSC provides a simplified mode to resolve the protein
folding problem. As a result, the astronomical number of folding
conformations can be easily stored into a database for protein folding.
Thus, the protein structure fingerprint approach made a significant
foundation to solve protein folding problem.

\textbf{Image visualization vs. alphabetic description.}

Due to complexity of protein structure, the protein structure
fingerprint provided the PFSC alphabetical description to probe a huge
number of protein data, especially it is suitable to study the protein
folding conformations with an astronomical number. The protein 3D
structure data are originally obtained by experimental measurements or
computational approaches, which pursue to display 3D image visualization
for protein structure. For single protein, its 3D structural image is
displayed according thousand lines of atomic coordinates in the protein
data file. Although a protein 3D structure is directly perceived through
the senses to understand the folding orientation in space, it is not
easily to illustrate the features of protein folding features. For
comparison of proteins, with structural superposition, the similarity is
quantified by the root-mean-square deviation (RMSD) as score.
Nevertheless, it does not provide any detail where and how are similar
or dissimilar between proteins, and artificial process severely affect
the outcome. So, it is hard to explain the similarity and dissimilarity
between proteins with 3D image visualization.4545Fitzkee NC, Fleming PJ,
Gong H, Panasik N, Street TO, Rose GD. Are proteins made from a limited
parts list? Trends Biochem Sci; 30:73--80, 2005.,4646Irving JA,
Whisstock JC, Lesk AM. Protein structural alignments and functional
genomics. Proteins; 42:378--382, 2001.,4747Sam V, Tai CH, Garnier J,
Gibrat JF, Lee B, Munson PJ. ROC and confusion analysis of structure
comparison methods identify the main causes of divergence from manual
protein classification. BMC Bioinform; 7:206, 2006.,4848Yang J. Complete
description of protein folding shapes for structural comparison{[}J{]}.
Proteomics Research Journal, 3(1):1-22, (2012).,4949Sarah A. Middleton,
Joseph Illuminati \& Junhyong Kim, Complete fold annotation of the human
proteome using a novel structural feature space, Scientific Reports
volume 7, Article number: 46321 (2017). Furthermore, it is almost
unimaginable to construct an astronomical number of 3D conformations for
a protein to probe the protein folding problem, and to involve with
billions of protein sequences even worse. However, one-dimensional PFSC
alphabetic string provided a useful protocol to overcome these obstacles
because it makes easily store and study a massive number of protein
conformations. The PFSC alphabetic representation does not only simplify
the description of protein conformation, but also it can align a large
number of folding conformations for comparison. With advance, the PFSC
alphabetic string covers the regular secondary fragments as well as the
tertiary fragments, so it became a valuable approach to study the
protein conformations with an astronomical number.

The alphabetic description has been adopted following development of
protein structure study. Except to label regular secondary motifs of
alpha helixes and beta strands, many different methods have been
developed trying to label protein conformation more detail with
alphabetic description. Some methods adopted more alphabetic letters to
distinguish secondary structure motifs in detail which specified the
patterns of hydrogen bonds and geometric criteria, such as C\selectlanguage{greek}α \selectlanguage{english}distances,
C\selectlanguage{greek}α \selectlanguage{english}angles, dihedral angles between C\selectlanguage{greek}α \selectlanguage{english}atoms, or a pairs of \selectlanguage{greek}ψ \selectlanguage{english}and \selectlanguage{greek}φ
\selectlanguage{english}dihedral angles around a C\selectlanguage{greek}α \selectlanguage{english}atom.5050Kabsch W, Sander C. Dictionary of
protein secondary structure: pattern recognition of hydrogen-bonded and
geometrical features. Biopolymers; 22:2577--2637, (1983).,5151Ridchards
FM, Kundrot CE. Identification of structural motifs from protein
coordinate data: secondary structure and first-level supersecondary
structure. Proteins; 3:71--84, (1988).,5252Frishman D, Argos P.
Knowledge-based protein secondary structure. Proteins; 23:566--579,
(1995).,5353Sklenar H, Etchebest C, Lavery R. Describing protein
structure: a general algorithm yielding complete helicoidal parameters
and aunique overall axis. Proteins; 6:46--60, (1989).,5454Labesse G,
Colloc'h N, Pothier J, Mornon JP. P-SEA: a new efficient assignment of
secondary structure from C alpha trace of proteins. Comput Appl Biosci;
13:3:291--295, (1997).,5555Martin J, Letellier G, Marin A, Taly JF, de
Brevern AG, Gibrat JF. Protein secondary structure assignment revisited:
a detailed analysis of different assignment methods. BMC Struct Biol;
5:17--34, (2005). Other methods identified the patterns of structural
segments with observations from a large number of structures in training
database, and extracted certain motifs as folding prototypes by
statistics adjustment and then labeled with alphabetic
letters.5656Fetrow JS, Palumbo MJ, Berg G. Patterns, structures, and
amino acid frequencies in structural building blocks, a protein
secondary structure classification scheme. Proteins; 27:249--271,
(1997).,5757Zhang X, Fetrow JS, Berg G. Design of an auto-associative
neural network with hidden layer activations that were used to
reclassify local protein structures. In: Crabb VJ, editor. Advances in
Protein Chemistry. San Diego, CA: Academic Press; pp 397--404
(1994).,5858Brevern AG, Etchebest C, Hazout S. Bayesian probabilistic
approach for predicting backbone structures in terms of protein blocks.
Proteins; 41:271--287, (2000).,5959Alexandre G, de Brevern1, Valadie\selectlanguage{ngerman}´ H,
Hazout S, Etchebest C. Extension of a local backbone description using a
structural alphabet: a new approach to the sequence-structure
relationship. Prot Sci; 11:2871--2886, (2002).,6060Fourrier L, Benros C,
Brevern AG. Use of a structural alphabet for analysis of short loops
connecting repetitive structures. BMC Bioinform; 5:58,
(2004).,6161Joseph A P, Srinivasan N, Brevern A G D. Improvement of
protein structure comparison using a structural alphabet{[}J{]}.
Biochimie, 93(9):1434, (2011). So far, most of alphabetic methods
adopted 9-16 letters to describe various folding protocols with
different lengths in fragments. Nevertheless, none of methods guarantee
to provide a complete coverage for all possible folding patterns due to
ignoring some of fragment motifs, such as irregular loops and coils or
uncommon folding shape with rare appearances in structures, etc.
However, the PFSC overcome the shortcomings, it provided a set of 27
alphabetical letters to cover all possible folds for successive 5 amino
acid residues, and a PFSC string describe the complete folding
conformation without gaps from N-terminus to C-terminus including
regular secondary fragments and irregular tertiary fragments.

The protein structure fingerprint can describe the folding conformations
with alphabetic description, no matter what the protein 3D structure is
known or unknown. For protein with known 3D structure, the folding shape
of each of 5 amino acid residues is assigned by one of PFSC letter
according the atomic coordinates, and then the conformation of entire
protein is expressed by a PFSC string. For protein without known 3D
structure, the comprehensive folding variations for a protein are able
simultaneously to be observed by the PFSC letters in PFVM with
impressiveness covering all at one glance. Also, an astronomical number
of folding conformations for a protein can be assembled with various
PFSC letters in PFVM. Furthermore, any PFSC string represents one of
folding conformations, and it can be conversely converted into 3D
structure.

The alphabetic letters provide a brief description for biological
structure in macromolecule system. The DNA polymer applies four letters
(C, G, A and T) to describe the backbone strand comprised of four
deoxyribonucleic acids in genetic code. The protein polymer applies 20
of amino acids with single letters to describe one-dimensional sequence.
Biological structure is embedded in assembly processes, from
one-dimensional DNA, mRNA to protein sequence until protein folding. In
the first step, the genetic information is stored in the DNA sequence
and transmitted through transcriptional and translated into
one-dimension protein sequence. In the second step, the protein is
folded from one-dimensional sequence to 3D structure for expressing the
vitality of life. To date, however, the knowledge and understanding of
protein folding lag far behind the DNA and protein sequences. The
protein structure fingerprint made a significant progress which applied
a set of 27 PFSC letters to describe protein folding. Thus, the PFSC
perfectly matched alphabetic description of DNA and protein sequence,
and it is possible to integrate the~huge~data of protein folding
conformations with DNA or mRNA sequence and protein sequence.

\textbf{Protein folding vs. the order of amino acids in sequence}

It is well known that the protein folding in principal depends on the
order of amino acids in sequence. Although researchers confirmed this
principal with many biological experiments, it lacks a systematical
depiction in bioinformatics aspect. Also, it is not easy to clearly
illustrate how the order of amino acids in sequence affects the folding
changes in protein. However, with a universal process, the PFVM
integrally displays the correlation between protein folding changes and
sequence variations. Generally, different protein sequences will have
different folding patterns in PFVM. The folding pattern difference is
presented in several aspects in PFVM even if only one amino acid was
substituted. The differences include the changes of the types of folding
shapes as well as the number of possible folding. Also, if one of amino
acid is substituted, it will not only cause PFSC letter changes in one
column, but a band of 5 columns in PFVM. These changes in PFVM well
demonstrated that the protein folding depended on the order of amino
acids in sequence.

The PFVM characteristically display the local folding variations along
the sequence. The numbers changes of local folding shapes display the
analogous fluctuation spectrum, and indicate some portions of protein
with more flexible while other portions with less flexible. The
fluctuation curves of numbers of local folding shapes for protein
PDCD1\_MOUSE and PDCD1\_HUMAN are shown in Figure 6. First, each curve
exposed how the folding flexibility following the order of amino acids
in sequence. Second, both curves are different because of the
differentiation of amino acids in sequences. At least, 4 locations in
curves (35-43, 78-91, 139-151 and 170-179 in sequence) have the opposite
tendency for the vibration of numbers of local folding variations. Thus,
a fluctuation curve from PFVM concretely indicates how the protein
folding relates the order of amino acid in sequence. Thus, the PFVM is a
useful tool to probe the protein mutation, protein differentiation,
protein design, protein prediction and protein misfolding etc.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image16/image16}

\end{center}
\end{figure}

Figure 6. The fluctuation curves of numbers of local folding shapes for
PDCD1\_MOUSE (red) and PDCD1\_HUMAN (blue) proteins. The horizontal axis
is the ruler of sequence, and vertical axis the number of local folding
variations.

\textbf{Disorder and flexibility in protein}

The nature of protein conformation in biological system is not static,
it is flexible following time being and condition change. The process
involves intermediate folding states and stable folding states during
their synthesis and degradation. These protein folding states may have
various half-lifetime, and keep reversible disorder-order transitions.
Also, the~intrinsically disordered folds indeed exist in some of
fragments or entire protein.11Dunker AK, Lawson JD, Brown CJ, Williams
RM, Romero P, Oh JS, Oldfield CJ, Campen AM, Ratliff CM, Hipps KW, Ausio
J, Nissen MS, Reeves R, Kang C, Kissinger CR, Bailey RW, Griswold MD,
Chiu W, Garner EC, Obradovic Z, ''Intrinsically disordered protein''.
Journal of Molecular Graphics \& Modelling. 19 (1): 26--59,
(2001).\textsuperscript{,}22Dyson HJ, Wright PE, ''Intrinsically
unstructured proteins and their functions''. Nature Reviews. Molecular
Cell Biology. 6 (3): 197--208, (2005)..\textsuperscript{,}33Dunker AK,
Silman I, Uversky VN, Sussman JL, ''Function and structure of inherently
disordered proteins''. Current Opinion in Structural Biology. 18 (6):
756--64, (2008).

The results from experimental measurement are actually affected by
intrinsically disorder and flexibility in protein. About two-thirds of
data in PDB do not have the complete 3D structures covering entire
sequence by X-ray crystallography measurement because of the unobserved
regions that frequently correspond to the disorder.44Le Gall T, Romero
PR, Cortese MS, Uversky VN,Dunker AK. Intrinsic disorder in the Protein
Data Bank. J Biomol Struct Dyn; 24:325-42, (2007). With nuclear magnetic
resonance (NMR) spectroscopy, it can reveal an ensemble of protein
structures with flexible dynamics or disorder states limited around a
static state. With transmission electron cryomicroscopy~(CryoTEM), it
can provide new insights for protein structure with large assemblies
which allow observation of protein structures in their native
environment at cryogenic temperature. With computational simulations, it
can obtain protein dynamics trajectories which reveal the disorder and
flexibility within protein structure. The protein folding occur in many
steps, and it may spend nearly 96\% time in some states,~~and also in
various intermediate states with minimum thermodynamic free
energies~in~energy landscape.55Heath Ecroyd; John A. Carver,
''Unraveling the mysteries of protein folding and misfolding''.~IUBMB
Life~(review).~60~(12): 769--774, (2008).~,66Robert B Best, ''Atomistic
molecular simulations of protein folding''.~Current Opinion in
Structural Biology~(review).~22~(1): 52--61, (2012).

Under physiological condition, the protein flexibility substantially
plays important role for biological functions. Even after self-assemble
into a native state with active function, some parts of proteins~may
remain folding variations.77Berg JM, Tymoczko JL,~Stryer L,~''3. Protein
Structure and Function''.~Biochemistry. San Francisco: W. H.
Freeman.~ISBN~0-7167-4684-0, (2002).~ The protein biological functions
associate with either the native stable structure or the dynamics~motion
in structure. It is undeniable that the protein structures with both
dynamic disorder conformations and stable conformations subsequently are
linked to important functions such as~allosteric regulation~and~enzyme
catalysis.88Bu Z, Callaway DJ, ''Proteins move! Protein dynamics and
long-range allostery in cell signaling''.~Advances in Protein Chemistry
and Structural Biology. Advances in Protein Chemistry and Structural
Biology.~83: 163--221, (2011).\textsuperscript{,}99Kamerlin SC, Warshel
A, ''At the dawn of the 21st century: Is dynamics the missing link for
understanding enzyme catalysis?''.~Proteins.~78~(6): 1339--75, (2010).~~
Although the so-called native conformation is essential to a specific
function in cell, the environmental factors, ligands and other proteins
may cause the native conformation into altered folds which may trigger
to act different biological function, such as active or inhibiting
function or toxic affection. This is actual process when a drug targets
proteins to make the conformation changes and to cause activation or
inhibition for protein function. Overall, a complete ensemble of
conformations really represents the nature of protein folding for
multiple functions in physiological condition or various environments.
It~is~significant~that the PFVM provides the comprehensive local folding
variations for discovery of various conformations in astronomical
number. Of course, the remaining question is how to distinguish local
folding variations in PFVM with association of function changes.
Ultimately, we need better experimental data and further data mining to
understand the role of each local folding variation in biological
functions.

\textbf{CONCLUSION}

The protein structure fingerprint provided a significant means to probe
the protein folding problem, especially overcame the obstacles how to
reveal and handle an astronomical number of folding conformations. The
folding shapes of 5 amino acid residues, as a universal element, are
comprehended, and then expanded over sequence to expose an astronomical
number of folding conformations. A set of 27 PFSC alphabetical letters
are represented as digital expression. The local folding variations in
PFVM really uncovered how the protein folds are correlated by the order
of amino acids in sequence. Also, it provides the prospect to construct
all possible conformations with astronomical number as well as to
predict the most probable conformations for a protein. These advantages
information may promote the research in protein structures, such as the
protein folding, protein~structure~prediction, intrinsically disordered
protein, protein mutation and protein design etc.

\textbf{DATA AVAILABILITY}

\begin{itemize}
\tightlist
\item
  All sequences are from UniProt database: https://www.uniprot.org/
\item
  All protein 3D structures are from PDB: https://www.rcsb.org/
\item
  The PFSC and PFVM can obtained from website http://www.micropht.com/
  or http://pfsc.considerin.net
\end{itemize}

\textbf{COMPETING INTERESTS STATEMENT}

We declare that we have no competing interests.

\textbf{ACKNOWLEDGEMENTS}

The protein structure fingerprint scheme has been developed by author
Jiaan Yang and published on ``Yang J, Comprehensive description of
protein structures using protein folding shape code. Proteins
2008;71.3:1497-1518''

\textbf{Supplementary Information:}

The supplementary information was stored in a file with rich text
format~(RTF), which hold longer sequence in same row and better display
the PFVM matrix. The name of supplementary file name is
``Supplementary-PFVM-Matrix-for 14 proteins.rtf''.

\textbf{Author Contributions}

G.W. involved input data preparation; W.X.C. involved output data
collection and analysis; X.F.Z. wrote the software code; Q.H, H.G, Q.Q,
X.J and L.Z. verified algebraic calculations; S.T.S. deployed software
code to web page; P.Z. tested and verified the code and final data; J.Y.
overall designed and managed the project and made the manuscript
preparation.

\textbf{Contact Information}

Correspondence and requests for materials should be addressed to J.Y.
(jyang@micropht.com)

\textbf{REFERENCE}

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