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Fundamental Thermodynamics of Protein Folding

Fundamental Thermodynamics of Protein Folding: Theory and Prediction
Table of Content
1.0 Introduction
2.0 Thermodynamic Theories of Protein Folding
3.0 Statistical Thermodynamics of Protein Folding
3.1 Hydrophobic Interaction
3.2 Hydrogen Bonds
3.3 Electrostatic Interaction
3.4 Van der Waals Forces
4.0 Protein Structure Prediction and Optimization
5.0 Conclusion and Future Prospective
6.0 References
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
Enthalpy change
Entropy change
Thermodynamic temperature
Gibbs free energy change
the free energy required to open a folded
protein with electrostatic interaction
the free energy required to open a folded
protein without electrostatic interaction
the free energy contribution of electrostatic
interaction in the folding process
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
The thermodynamic hypothesis proposed by Anfinsen states the native protein always
has the lowest energy level which is also graphically represented by the energy landscape,
becoming the basic theory of ab initio structure prediction. In addition, on the basis of
traditional thermodynamics, statistical thermodynamics provides a clearer way to calculate and
analyze thermodynamic properties of protein folding process from a macro perspective. This
report focuses on the folding process explanation in both classic thermodynamic and statistical
thermodynamic theories, as well as the protein structure prediction. Through searching this
process thermodynamically, the fundamental principle will be known. However, protein
folding is a so complexed process, new conceptual breakthroughs will be required to obtain
further progress.
1.0 Introduction
As the fundamental components for almost every living system, there are more than
100,000 kinds of proteins performing different functions in the human body like enzymes,
regulatory proteins and immunoglobulin. Such functional difference is mainly determined by
the amounts of amino acids and their sequencing, namely, the primary structure of proteins.
Generally, proteins are considered to have four hierarchical structures [1]. The primary
structure is the sequence of different types of amino acids on the polypeptide chain. The
e i he eg la
e fo med locall b
he ol
e ide chain,
ch a
heli and -sheet. Tertiary structure is a 3D structure formed by the close arrangement of
secondary structures in space. The quaternary structure refers to the compound molecule
formed by the interaction between different polypeptide chains. After the unfolded protein
undergoes a complex folding process, the complex structure of the natural protein formed has
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
different functions and undertakes different tasks in the organism.
This delicate construction, as the machine of life, mediates most of the processes taking
place in the living cells. The unique features for the most important polymeric molecules,
which distinguish from all other polymers with a broad range of properties and functions, are
their peculiar structure configuration. It is not difficult to see that protein is an important bearer
of life activities. Therefore, it is important to research the folding mechanism of proteins.
The significance of studying the principles of protein folding is threefold: First, it can
help to find the reason for protein folding. Second, it can help to understand the structure and
function of proteins more clearly. Third, it can provide a basis for the design of new proteins
and targeted drugs. Fourth, despite most processes in living cells proceed in disequilibrium
situations, measuring thermodynamic properties of the proteins to analyze is a more efficient
and reliable method, which can make an essential contribution to biology.
It is proven extensively that the folding process from polypeptide to protein is under
thermodynamic and kinetic control [2]. In this report, main interests will fall on analysing this
typical process in a thermodynamic view and explaining why the processes proceed the way
they do.
The first part mainly devotes to the theories. Generally, the folding and construction
process is sophisticated and complex. Only the first three stages will be discussed in this article,
which are on account for thermodynamics and molecular kinetics. Moreover, analysing the
typical process and explaining the mechanism in a thermodynamic view are at the core of this
report. Here, the theories will be listed in both kinetic and thermal views, and it is believed
both types of control are active simultaneously. To be more specific, two main theories,
Anfin en
and ene g land ca e fo
o ein
ill be anal ed follo ing o cla if
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
construction process [3]. Eventually, practical experiments in protein constructions will be
provided for further explanation.
In the second part, the extensive new approach currently to analyze the protein folding
process, statistical thermodynamics, is introduced generally. Specifically, the effects of
hydrophobic interaction, hydrogen bonding, electrostatic interaction, and van der Waals force
on the stability of the protein structure and the free energy of the folding process will be
analyzed in detail.
In the third part, the ab initio structure prediction will be introduced and also three kinds
of optimization methods.
2.0 Thermodynamic Theories of Protein Folding
It is proven that there are three levels in the process of a protein folding process (The
fourth is about the interaction between chains, which will not be covered by this article.). The
first level is the sequence of the peptide, and the second is some basic tortuosity and distortion.
These two spatial structures are believed to be controlled by thermodynamics. Thus, the
essential question leaves to how the polypeptide in the first level controls protein dimensional
folding for the second level thermodynamically.
Two thermodynamic ways have been put forward for constructing protein folding
models. The first way predicts the native protein molecules conformation in the lowest
thermodynamic energy state are the most stable configurations. On account of all interaction
forces among molecules and the interaction between the whole protein particles and solvent,
simulate the protein natural spatial structure with minimum energy on the theory of molecular
mechanics. The other one focuses on the comparison between thermodynamic properties
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
statistics from experimental measurement and current protein database, thereby finding the
regulation and constructing dimensional models. The structure forecast is based on protein
homology and probing the conformation periodically, getting the structures level by level. To
be specific, the first level statistic will be used in constructing the second level of configuration,
following the more complex dimensional protein structures. With the regulation attained in the
process and the feasible models concluded from the experiment, the unreasonable structures
could be abandoned, and further modification are accessible with theory of the lowest energy
level of the molecules. However, the first way is extremely difficult in the mathematical field
while the second one has some accuracy limitations.
In 1973, Anfinsen first analysed the connection between peptide sequence and the
secondary structure for polypeptides chain [4]. In the studies on ribonuclease, he and his group
found that the denaturation ribonuclease had the ability to renature spontaneously.
Subsequently in their vivo and vitro experiments, they found out a kind of enzyme in cells
which catalysed the folding process by discovering thermodynamic unstable positions. From
the above experiments Anfinsen came out two conclusions. One was that the native protein had
the lowest free energy under the constant physical conditions included temperature, pressure,
pH and so on, so once a protein did not stay in a steady lowest way, it would decrease its free
energy automatically in order to achieve the thermodynamic favourable position, which was
called the Thermodynamic Hypothesis [4]. Another proven in his experiment was that all
information needed for polypeptides to fold into basic secondary structure such as 𝛼-helices
and 𝛽 -strands and their order contribute into the complicated structure entirely contained
within the amino acid sequence [5]. After folding into the secondary conformation, the energy
level will decrease compared to the unfold chains.
In the following years, many experiments and models about Anfinsen Dogma had been
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
put forward by scientists. In 1993, Kolinski and Skolnick successfully folded model helical
proteins [6]. Among their final cultured proteins, the energies of those unsuccessful folding
simulations which presented incorrect structures were about 50-60 kBT higher than the correct
ones had.
However, it is still shocking to have an exception, which after the folding process the
conformation acquires a higher energy state than the former (Figure 1A and 1B). Chaperones
are discovered to explain this unique phenomenon. Classical chaperones were proven to exist
in many living cells, yet they only contribute to the folding efficiency or accelerate the process
rather than alter the reaction path. Highly specific steric chaperons were discovered, which
have astonishing abilities to transfer some converting information to the native protein
configurations, meaning the barrier between native and partially folded state will decrease
which enable the folding process to proceed in an energy raising direction (Figure 1C). Though
ome aid he e i ing of hi
e of
e ic cha e one
iola e
he Anfin en
la , i
should be counted into an exception, and further validation is required at a deeper level.
Figure.1 (AοΌ‰ A normal folding process energy changing diagram. (BοΌ‰ Diagram for folded protein
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
which embraces a higher energy level than dissociate. (CοΌ‰ Gibbs free energy level changing diagram with
the use of steric chaperones [5].
Tho gh Anfin en
inci le ha gi en i e o
ildl do b ed d e o i
ligh conflic
towards thermodynamics, it is acknowledged by most that the folding process belongs to an
irreversible thermodynamic process. However, for the lack of criteria and standard
measurements, the scientists can only construct their models relying on both theory and
experiment experience. In this circumstance, Frauenfelder first came up with the concept of the
energy landscape for explanation. The landscape is based on the second thermodynamic
principle emphasizing the total free energy of protein and surroundings decrease while folding,
yet somehow protein energy level will not always drop during the process [7]. In Onuchic
perspective, polypeptide chains firstly collapse on their configurations to form a compact
structure, which is only similar in shape with an active protein structure, and then rebuild their
most stable conformation through steric stretching, twisting, transient and bonding. The
process above leads to a funnel ha ed o m l i f nnel ha ed land ca e, he e he Anfin en
thermodynamic principle could narrowly explain the funnel shaped diagram, yet the multi one
is really beyond its reach. For the multi funnel shaped landscape, a new question occurs, which
is among all the equivalent funnels why the native funnel is selected for the structure. The
explanation is the transient forces need to push a given 𝛼-amino acid sequence into the native
funnel comes from vibrational excited states, also known as the VES hypothesis. The VES
hypothesis triggers the transient forces that constitute the first step in protein folding and
function [8].
To validate the essential ideal that the conformations construct themselves to the global
minimum Gibbs free energy structures, three different configurations of four types of protein
were measured thermodynamically in Cruzeiro et al experiment. By comparing the energy
landscapes obtained, it was proven that in a multi-funnel Gibbs energy landscape, each funnel
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
represents an average structure with different energy from the native, and each structure in a
different funnel is theoretically attainable (Figure 2). In spite of the diverse structures in thermal
possibility, the structure is not solely decided by Gibbs energy minimization. It was proposed
by Levinthal in early 1968 that there is a first funnel selecting step before the folding process
based on the non-equilibrium kinetic mechanism. And now such a method is proven to be the
VES hypothesis mentioned above.
Figure.2. A funnel shape energy landscape example for 𝛼 -helix protein folding process. Measured by
effective energy and different conformation as well as entropy [8].
There is no doubt at all that the peptide sequence decides the landscape. Debayan et al
experiment aiming to link the peptides encoding with Gibbs energy of the process is based on
the transition from an 𝛼-helix to a 𝛽-harpin [9]. The polypeptide chain was characterized by
using Discrete Path Sampling (DPS) technique. The contrast was designed to encode with DP3
from DP5 in peptide sequences, mapped out by DPS. And the multi-funnel energy landscape
of the DP3 set was reshaped after recording the sequences (Figure 3). By comparing the 𝛼helix and the 𝛽 -harpin configurations after changing the peptide sequences, both the
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
thermodynamic and kinetic properties are changed. In microscopic view, the molecular
mechanism and transient feature remained unchanged, meaning the adjustment of the energy
landscape might be resulted by changing of the key hydrogen-binding during the process [9].
Figure.3. (a) Multi-funnel energy landscape of the 𝛼-helix and the 𝛽-harpin of DP5 sequences in 300K. Blue
branches represent the helical conformations, while the red represent the hairpin conformations. Other partial
structures are also shown on the graph; (b) Multi-funnel energy landscape of the 𝛼-helix and the 𝛽-harpin of
DP3 sequences in 300K. Blue branches represent the helical conformations, while the red represent the
hairpin conformations. Other partial structures are also shown on the graph [9].
Next phase for the folding process is a kinetic dominant stage, where the secondary
structure begins to fold into more complex steric conformation. The process mainly proceeds
in externalities, where the pressure, temperature and acidity are uncertain. And due to the
kinetic view, the process can be reversible depending on the external condition. For this part
of the process, the equilibrium has to be reached. It is known that all systems evolve towards
equilibrium state, and an isolated system characterizes by maximum entropy. In contrast, living
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
systems seem to violate the truth with proceeding to increase order of the system, and never
reach equilibrium. However, universe entropy must increase, which feeds the entropy
negatively to the biology subsystem to keep the latter evade from equilibrium. Another word,
the subsystem might demonstrate a trend away from equilibrium and decrease in entropy, but
the things will keep towards entropy increase overall [10]. The equilibrium might collapse with
changing of the external condition, such as changing pressure, or changing pH. The third level
structure might damage and disable basic living function due to the equilibrium changing called
denature for the protein conformation. The damaged structure might fold back again if the
external condition falls back again. However, if the deeper configuration is damaged, generally
the secondary structure, the protein could not go back.
3.0 Statistical Thermodynamics of Protein Folding
It is difficult to describe the folding process of a single protein molecule. First, a single
protein molecule keeps doing random thermal motion and has different structural states, which
is hard to describe its trajectory and precise state. Second, protein molecules do not just have
one unique spatial structure but are ensembles of interconnected and interconvertible
microscopic states. For example, the structure of natural protein crystals observed by NMR is
the average ensembles of different microscopic states rather than a stable structure [11].
Therefore, it is of great significance to study the collection of microscopic states composed of
natural proteins from the perspective of statistical thermodynamics.
Protein folding is mainly affected by factors, including hydrophobic interaction,
hydrogen bonding, electrostatic force, and van der Waals force. The second part of this article
will discuss how these four factors influence the stability of the protein and the free energy of
the folding process.
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
3.1 Hydrophobic Interaction
Among the twenty common amino acids in nature, they can be divided into two
categories according to their hydrophilicity and hydrophobicity. Nine amino acids are
hydrophobic amino acids, and the remaining eleven amino acids are hydrophilic amino acids.
There are both hydrophobic residues and hydrophilic residues in an unfolded protein chain.
The side chain which ends of hydrophobic residues are non-polar hydrocarbon groups, while
the carbonyl and amino groups in the main chain are hydrophilic groups. Under the combined
action of the hydrophobic groups and the hydrophilic groups, the hydrophobic groups will
gather inside the protein to form a hydrophobic core, while the hydrophilic groups will form
an approximately spherical coating on the protein surface. Therefore, natural protein can be
regarded as an approximately spherical structure with a hydrophobic core inside and a
hydrophilic coating outside. Studies have shown that hydrophobic interaction is an important
driving force for the rapid inward collapse in proteins folding process [12].
As the destruction of the tight hydrophobic core requires energy, the process of the
hydrophobic groups in an unfolded protein collapsing inward to form the hydrophobic core is
an exothermic process, which means the enthalpy change is negative.
As the number of spatial conformations of an unfolded protein chain is greater than
natural proteins, the process of protein folding is an entropy reduction process, which means
the entropy change is negative.
The relationship between the change of free energy and the change of enthalpy and the
change of entropy is:
G= H-T S
Experiments and studies have confirmed that under the influence of hydrophobic effect,
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
the reduction of enthalpy is the main influencing factor. The free energy of this process is
negative, which shows that protein collapse is a spontaneous process [12].
3.2 Hydrogen Bonds
A hydrogen bond is an interaction formed between a hydrogen atom and an atom with
great electronegativity, which usually means an oxygen atom or a nitrogen atom. Hydrogen
bonds are directional. For proteins, hydrogen bonds can only exist when the carbonyl group
and amino group meet a certain angle.
Hydrogen bonds are widely found in proteins. In -helix, a hydrogen bond can be
formed between the upper circle and the lower circle of the helix. In -sheet, a hydrogen bond
can be formed between one strand and another antiparallel strand. As the hydrogen bonds are
regularly present between residues in -helix and -sheet, they can make the protein in a stable
state. The hydrogen bonds are the key force in the formation of secondary structure.
The formation of hydrogen bonds between residues and residues will release a lot of
energy, resulting in a decrease in enthalpy.
Shirley et al. studied the hydrogen bond changes of ribonuclease T1 mutants [13]. They
pointed out that the average hydrogen bond contribution to structural stability was about 1.3kcal/mol. Considering that there are many hydrogen bonds in one protein, it can be seen that
the formation of hydrogen bonds plays a vital role in the stability of protein structure.
3.3 Electrostatic Interaction
Among eleven hydrophilic amino acids, there are three positively charged amino acids,
including arginine, histidine, and lysine, and two negatively charged amino acids, including
aspartic acid and glutamic acid. When these charged residues are close to each other, they will
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
generate electrostatic interaction between each other, also known as salt bridge. The
electrostatic interaction makes the charged residues attract each other and become closer, which
reduces free energy and makes the protein structure more stable.
In addition, as the charged residues are hydrophilic, they are usually located in the polar
environment outside the protein. However, in the folding process, it is inevitable that some of
the charged residues will be involved into the hydrophobic core, and this process is called
desolvation of charges. The accumulation of charges in the hydrophobic core makes protein
structure more unstable, causing an increase in enthalpy. In other words, the desolvation
process will increase the free energy and make the protein structure more unstable [14].
If π›₯πΊπ‘ˆ,𝑠𝑏 is used to denote the free energy required to open a folded protein with
electrostatic interaction, and π›₯πΊπ‘ˆ,π‘›π‘œπ‘›
is used to denote the free energy required to open a
folded protein without electrostatic interaction, then the free energy contribution π›₯π›₯πΊπ‘ˆ of
electrostatic interaction to the folding process is:
If π›₯π›₯πΊπ‘ˆ >0, the electrostatic interaction makes the structure of the protein more stable.
Conversely, the electrostatic interaction makes the protein structure more unstable. Since the
interaction between salt bridges and the desolvation of charges are close, the effect of
electrostatic force on the stability of protein structure is in a critical state. In the specific
analysis, whether static electricity will make the protein structure more stable or unstable
depends on which factor plays the leading role.
3.4 Van der Waals Forces
Van der Waals force is the force between molecules, including gravitation and
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
repulsion. Due to the diversity of protein structures, the influence of van der Waals forces on
the stability of protein structures is also complicated. Because van der Waals forces can attract
side chains closer to each other, the tertiary structure of natural protein can be tighter. In
addition, studies have shown that the attraction of van der Waals force can bring the amino
group and the carbonyl group close, thereby enhancing the hydrogen bond force, and reducing
the free energy of the protein, making the system more stable [15].
Since van der Waals forces exist widely between molecules and are affected by the
complex spatial structure of different proteins, it is still challenging to quantitatively analyze
the effects of van der Waals forces on protein stability and free energy change during the
folding process.
4.0 Protein Structure Prediction and Optimization
At the 18th Critical Assessment of Protein Structure Prediction in 2018, the Alphafold
from Deepmind
edic ed 25 kind
o ein
cham ion hi b a ignifican ad an age. Thi no onl in od ce
cce f ll
eo le
on he
i ion in o a dee
study field, but also leads to a rethinking of traditional protein structure algorithms.
At present, there are three main kinds of prediction methods, the homologous modelling,
folds recognition and ab initio structure prediction. For the first two methods, they rely on the
resolved protein structures. By comparing the protein structure with the templates from the
protein data bank, the target model could be constructed. But the number of protein structures
in the Bank was only 100,000. And the known protein sequences had been about 90 million by
2015 [17]. The huge gap makes the template-based methods invalid under most conditions
(Figure 4). In this case, we need to construct 3D models from scratch, which is called ab initio
structure prediction.
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
Figure.4. The amounts of known protein sequences and solved protein structures from 1995 to 2015 are
shown respectively [17].
Ab initio protein structure prediction which has been developed for more than 60 years
is still a promising prediction method and it is regarded as the holy grail of molecular biology
[16]. Compared to the homologous modelling and folds recognition, the ab-initio method is an
ideal method to predict the protein structure without the known protein structures but only
relying on the amino sequences. Because of template-free, it is the hardest one among protein
structure prediction approaches. But it could be much helpful for people to understand the
fundamental protein folding mechanism and how a chain of polypeptides could change into a
specific function protein [17].
The theory basis of the initio method is the thermodynamic hypothesis that the native
protein corresponds to the global minimum free energy which was proposed by Anfinsen in
1973 [4]. For the ab-initio method, the basic protocol to predict protein structures is to search
for the conformations of an appropriate potential energy which could lead to the prediction of
native folds. After the folds have been recognized or predicted, the predicted structures will be
assessed of the quality to determine whether the structure is rejected or selected. But there are
two main constraints to the successful implementation of the ab-initio method. One is the lack
of an effective potential function which could distinguish the native conformation from the
non-native conformation of proteins so that the global minimum energy function corresponds
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
to the native proteins. Secondly, there are an astronomical number of local minima on the
potential surface and it is hard to sample it efficiently.
In 1983, Kirkpatrick was first inspired by the solid annealing process that according to
the Boltzmann probability, when the temperature is lower, the atoms would collapse into the
lowest-energy state and created the simulated annealing (SA) as a powerful optimization
method which is probably the most common used method in protein structure prediction by far
[18]. In this method, an applied algorithm generates a series of conformations following the
Boltzman energy distribution at a given temperature. It performs a high temperature simulation,
followed by a series of simulations based on the temperature reduction plan to find the ground
As an improvement of SA, conformational space annealing which could search for a
larger number of low-energy families was introduced by Baker Laboratory [19]. In this method,
they first build a bank containing a preassigned number of random conformations which will
subsequently be energy-minimized. Then several dissimilar ones will be selected as seeds to
be modified. Through reducing the distance between conformations and updating the bank, the
lower-energy locations will be found out. After assessing, the native-like structure will be
determined (Figure 5).
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
Figure.5. Schematic diagram of the conformational space annealing.
There are also many other conformational search methods, such as the entropic
ensemble which is based on the system entropy at initial temperature in order to provide the
entropy estimation of a larger system for Monte Carlo simulations [20].
5.0 Conclusion and future prospective
The basic thermodynamic mechanism for each stage of the protein folding process has
been discussed above. Viewing the process thermodynamically not only provides a new
concept to explain and understand the protein folding, but also enables protein configuration
forecast. Actually, protein folding is an extremely complex process which covers dozens of
subjects, including thermodynamics, biology, chemistry, molecular kinetic and so on, and it is
impossible to deduce the process only on thermodynamics, while it still gives an overall
concept and explanation fundamentally.
The multi-funnel landscape investigates the
possibility within polypeptides folding by Gibbs energy for native structure, or uncommon
configurations. Combining Anfinsen's law and the energy landscape explains most of protein
structure. However, exceptions occur with which the folding could not act solely by Gibbs
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
energy minimization, and such phenomenon could be considered by the VES hypothesis, which
clarified deterministic force exists in first structure level and decides which specific
conformation to go. The regret is that the further mechanism for the hypothesis is still vague.
And now the experiment by Debayan et al proved that not just the conformation is decided by
peptide sequence but also the whole energy landscape in primary level. The last phase in
process is more about kinetic perspective, where the secondary structure, literally elementary
protein configuration with some basic spatial structures, fold into more complex protein
structure reversibly. And equilibrium exists and controls the process in this stage. And this
report mainly focuses on the principle for the first two levels of protein construction, as the
third is in the scope of molecular kinetic view.
However, limitations still exist. Unfold or fold process for some extreme large or
complex protein structure under certain temperature irreversibly is still difficult to be captured
by current models [21]. Also,the energy landscape could not reveal every type of protein
folding process, furthermore some of them are even not a funnel shape, which are far more
beyond current models and simulations.
Studies have shown that in the process of protein folding, hydrophobic interaction plays
the most important driving role. In addition, hydrogen bonds, electrostatic forces, and van der
Waals forces also affect the stability of the protein and the free energy of the folding process.
The research results help people understand the driving force of protein folding more deeply
and lays the foundation for further research on protein folding.
With the increase of successful genome sequencing projects, more and more amino
sequences have been known. But one essential approach to understand their functions is still
the protein structure information. Ab initio protein structure prediction method earned lots of
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
controversies during the past decades. Based on the thermodynamic hypothesis, the initio
method put forward higher requirements on optimization algorithms and effective energy
functions which were used to be hard to achieve. However, the success of AI Deep Study and
more advanced computing technology have been gradually radiating new vitality of the
traditional structure prediction. There is no doubt that a huge step further in protein structure
prediction will occur in the near future through combining the states of the art technologies and
the traditional thermodynamics.
Fundamental Thermodynamics of Protein Folding: Theory and Prediction
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Fundamental Thermodynamics of Protein Folding: Theory and Prediction
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