Statistical potential

Possible features to which a pseudo-energy can be assigned include:

• interatomic distances,
• torsion angles,
• solvent exposure,
• or hydrogen bond geometry.

The classic application is, however, based on pairwise amino acid contacts or distances, thus producing statistical interatomic potentials. For pairwise amino acid contacts, a statistical potential is formulated as an interaction matrix that assigns a weight or energy value to each possible pair of standard amino acids. The energy of a particular structural model is then the combined energy of all pairwise contacts (defined as two amino acids within a certain distance of each other) in the structure. The energies are determined using statistics on amino acid contacts in a database of known protein structures (obtained from the PDB).

Many textbooks present the statistical PMFs as proposed by Sippl as a simple consequence of the Boltzmann distribution, as applied to pairwise distances between amino acids. This is incorrect, but a useful start to introduce the construction of the potential in practice.

The Boltzmann distribution applied to a specific pair of amino acids,

is given by:

:$$

P\left(r\right)=\frac{1}{Z}e^{-\frac{F\left(r\right)}{kT}}

where $r$ is the distance, $k$ is the Boltzmann constant, $T$ is

the temperature and $Z$ is the partition function, with

:$$

Z=\int e^{-\frac{F(r)}{kT}}dr

The quantity $F(r)$ is the free energy assigned to the pairwise system.

Simple rearrangement results in the inverse Boltzmann formula,

which expresses the free energy $F(r)$ as a function of $P(r)$:

:$$

F\left(r\right)=-kT\ln P\left(r\right)-kT\ln Z

To construct a PMF, one then introduces a so-called reference state with a corresponding distribution $Q\_\{R\}$ and partition function

$Z\_\{R\}$, and calculates the following free energy difference:

:$$

\Delta F\left(r\right)=-kT\ln\frac{P\left(r\right)}{Q_{R}\left(r\right)}-kT\ln\frac{Z}{Z_{R}}

The reference state typically results from a hypothetical

system in which the specific interactions between the amino acids

are absent. The second term involving $Z$ and

$Z\_\{R\}$ can be ignored, as it is a constant.

In practice, $P(r)$ is estimated from the database of known protein

structures, while $Q\_\{R\}(r)$ typically results from calculations

or simulations. For example, $P(r)$ could be the conditional probability

of finding the $C\backslash beta$ atoms of a valine and a serine at a given

distance $r$ from each other, giving rise to the free energy difference

$\backslash Delta\; F$. The total free energy difference of a protein,

$\backslash Delta\; F\_\{\backslash textrm\{T\}\}$, is then claimed to be the sum

of all the pairwise free energies:

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where the sum runs over all amino acid pairs $a\_\{i\},a\_\{j\}$

(with

Intuitively, it is clear that a low value for $\backslash Delta\; F\_\{\backslash textrm\{T\}\}$ indicates

that the set of distances in a structure is more likely in proteins than

in the reference state. However, the physical meaning of these statistical PMFs has

been widely disputed, since their introduction.{{cite journal |vauthors=Koppensteiner WA, Sippl MJ | year = 1998 | title = Knowledge-based potentials–back to the roots | journal = Biochemistry Mosc. | volume = 63 | issue = 3| pages = 247–252 | pmid = 9526121 }}{{cite journal | author = Shortle D | year = 2003 | title = Propensities, probabilities, and the Boltzmann hypothesis | journal = Protein Sci | volume = 12 | issue = 6| pages = 1298–1302 | doi=10.1110/ps.0306903| pmid = 12761401 | pmc = 2323900}} The main issues are:

1. The wrong interpretation of this "potential" as a true, physically valid potential of mean force;
2. The nature of the so-called reference state and its optimal formulation;
3. The validity of generalizations beyond pairwise distances.

In response to the issue regarding the physical validity, the first justification of statistical PMFs was attempted by Sippl.{{cite journal |vauthors=Sippl MJ, Ortner M, Jaritz M, Lackner P, Flockner H | year = 1996 | title = Helmholtz free energies of atom pair interactions in proteins | journal = Fold Des | volume = 1 | issue = 4| pages = 289–98 | doi=10.1016/s1359-0278(96)00042-9| pmid = 9079391 | doi-access = }} It was based on an analogy with the statistical physics of liquids. For liquids, the potential of mean force is related to the radial distribution function $g(r)$, which is given by:Chandler D (1987) Introduction to Modern Statistical Mechanics. New York: Oxford University Press, USA.

:$$

g(r)=\frac{P(r)}{Q_{R}(r)}

where $P(r)$ and $Q\_\{R\}(r)$ are the respective probabilities of

finding two particles at a distance $r$ from each other in the liquid

and in the reference state. For liquids, the reference state

is clearly defined; it corresponds to the ideal gas, consisting of

non-interacting particles. The two-particle potential of mean force

$W(r)$ is related to $g(r)$ by:

:$$

W(r)=-kT\log g(r)=-kT\log\frac{P(r)}{Q_{R}(r)}

According to the reversible work theorem, the two-particle

potential of mean force $W(r)$ is the reversible work required to

bring two particles in the liquid from infinite separation to a distance

$r$ from each other.

Sippl justified the use of statistical PMFs—a few years after he introduced

them for use in protein structure prediction—by

appealing to the analogy with the reversible work theorem for liquids. For liquids, $g(r)$ can be experimentally measured

using small angle X-ray scattering; for proteins, $P(r)$ is obtained

from the set of known protein structures, as explained in the previous

section. However, as Ben-Naim wrote in a publication on the subject:

[...] the quantities, referred to as "statistical potentials," "structure

based potentials," or "pair potentials of mean force", as derived from

the protein data bank (PDB), are neither "potentials" nor "potentials of

mean force," in the ordinary sense as used in the literature on

liquids and solutions.

Moreover, this analogy does not solve the issue of how to specify a suitable reference state for proteins.

In the mid-2000s, authors started to combine multiple statistical potentials, derived from different structural features, into composite scores.{{Cite journal|last1=Eramian|first1=David|last2=Shen|first2=Min-yi|last3=Devos|first3=Damien|last4=Melo|first4=Francisco|last5=Sali|first5=Andrej|last6=Marti-Renom|first6=Marc|date=2006|title=A composite score for predicting errors in protein structure models|journal=Protein Science|volume=15|issue=7|pages=1653–1666|doi=10.1110/ps.062095806|pmc=2242555|pmid=16751606}} For that purpose, they used machine learning techniques, such as support vector machines (SVMs). Probabilistic neural networks (PNNs) have also been applied for the training of a position-specific distance-dependent statistical potential.{{Cite journal|last1=Zhao|first1=Feng|last2=Xu|first2=Jinbo|date=2012|title=A Position-Specific Distance-Dependent Statistical Potential for Protein Structure and Functional Study|journal=Structure|volume=20|issue=6|pages=1118–1126|doi=10.1016/j.str.2012.04.003|pmc=3372698|pmid=22608968}} In 2016, the DeepMind artificial intelligence research laboratory started to apply deep learning techniques to the development of a torsion- and distance-dependent statistical potential.{{cite journal |vauthors=Senior AW, Evans R, Jumper J, etal |title=Improved protein structure prediction using potentials from deep learning|journal=Nature|volume=577 |issue=7792 | pages=706–710 |year=2020 |pmid=31942072|doi=10.1038/s41586-019-1923-7|bibcode=2020Natur.577..706S|s2cid=210221987|url=https://discovery.ucl.ac.uk/id/eprint/10089234/1/343019_3_art_0_py4t4l_convrt.pdf}} The resulting method, named AlphaFold, won the 13th Critical Assessment of Techniques for Protein Structure Prediction (CASP) by correctly predicting the most accurate structure for 25 out of 43 free modelling domains.

Baker and co-workers {{cite journal |vauthors=Simons KT, Kooperberg C, Huang E, Baker D | year = 1997 | title = Assembly of protein tertiary structures from fragments with similar local sequences using simulated annealing and Bayesian scoring functions | journal = J Mol Biol | volume = 268 | issue = 1| pages = 209–225 | doi=10.1006/jmbi.1997.0959| pmid = 9149153 | citeseerx = 10.1.1.579.5647 }} justified statistical PMFs from a

Bayesian point of view and used these insights in the construction of

the coarse grained ROSETTA energy function. According

to Bayesian probability calculus, the conditional probability $P(X\backslash mid$

A) of a structure $X$, given the amino acid sequence $A$, can be

written as:

:$$

P\left(X\mid A\right)=\frac{P\left(A\mid

X\right)P\left(X\right)}{P\left(A\right)}\propto P\left(A\mid

X\right)P\left(X\right)

$P(X\backslash mid\; A)$ is proportional to the product of

the likelihood $P\backslash left(A\backslash mid\; X\backslash right)$ times the prior

$P\backslash left(X\backslash right)$. By assuming that the likelihood can be approximated

as a product of pairwise probabilities, and applying Bayes' theorem, the

likelihood can be written as:

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where the product runs over all amino acid pairs $a\_\{i\},a\_\{j\}$ (with

Obviously, the negative of the logarithm of the expression

has the same functional form as the classic

pairwise distance statistical PMFs, with the denominator playing the role of the

reference state. This explanation has two shortcomings: it relies on the unfounded assumption the likelihood can be expressed

as a product of pairwise probabilities, and it is purely qualitative.

Hamelryck and co-workers later gave a quantitative explanation for the statistical potentials, according to which they approximate a form of probabilistic reasoning due to Richard Jeffrey and named probability kinematics. This variant of Bayesian thinking (sometimes called "Jeffrey conditioning") allows updating a prior distribution based on new information on the probabilities of the elements of a partition on the support of the prior. From this point of view, (i) it is not necessary to assume that the database of protein structures—used to build the potentials—follows a Boltzmann distribution, (ii) statistical potentials generalize readily beyond pairwise differences, and (iii) the reference ratio is determined by the prior distribution.

Image:ratio reference method.svg. In order to obtain a complete description of protein structure, one also needs a probability distribution $P(Y)$ that describes nonlocal aspects, such as hydrogen bonding. $P(Y)$ is typically obtained from a set of solved protein structures from the PDB (left). In order to combine $Q(X)$ with $P(Y)$ in a meaningful way, one needs the reference ratio expression (bottom), which takes the signal in $Q(X)$ with respect to $Y$ into account.]]

Expressions that resemble statistical PMFs naturally result from the application of

probability theory to solve a fundamental problem that arises in protein

structure prediction: how to improve an imperfect probability

distribution $Q(X)$ over a first variable $X$ using a probability

distribution $P(Y)$ over a second variable $Y$, with $Y=f(X)$. Typically, $X$ and $Y$ are fine and coarse grained variables, respectively. For example, $Q(X)$ could concern

the local structure of the protein, while $P(Y)$ could concern the pairwise distances between the amino acids. In that case, $X$ could for example be a vector of dihedral angles that specifies all atom positions (assuming ideal bond lengths and angles).

In order to combine the two distributions, such that the local structure will be distributed according to $Q(X)$, while

the pairwise distances will be distributed according to $P(Y)$, the following expression is needed:

:$$

P(X,Y)=\frac{P(Y)}{Q(Y)}Q(X)

where $Q(Y)$ is the distribution over $Y$ implied by $Q(X)$. The ratio in the expression corresponds

to the PMF. Typically, $Q(X)$ is brought in by sampling (typically from a fragment library), and not explicitly evaluated; the ratio, which in contrast is explicitly evaluated, corresponds to Sippl's PMF. This explanation is quantitive, and allows the generalization of statistical PMFs from pairwise distances to arbitrary coarse grained variables. It also

provides a rigorous definition of the reference state, which is implied by $Q(X)$. Conventional applications of pairwise distance statistical PMFs usually lack two

necessary features to make them fully rigorous: the use of a proper probability distribution over pairwise distances in proteins, and the recognition that the reference state is rigorously

defined by $Q(X)$.

Statistical potentials are used as energy functions in the assessment of an ensemble of structural models produced by homology modeling or protein threading. Many differently parameterized statistical potentials have been shown to successfully identify the native state structure from an ensemble of decoy or non-native structures.{{cite journal |author=Lam SD, Das S, Sillitoe I, Orengo C |title=An overview of comparative modelling and resources dedicated to large-scale modelling of genome sequences |journal= Acta Crystallogr D |volume=73 |issue=8 |pages=628–640 |year=2017 |pmid= 28777078 |doi=10.1107/S2059798317008920 |pmc=5571743|bibcode=2017AcCrD..73..628L }} Statistical potentials are not only used for protein structure prediction, but also for modelling the protein folding pathway.{{cite journal |author=Kmiecik S and Kolinski A |title=Characterization of protein-folding pathways by reduced-space modeling |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=104 |issue=30 |pages=12330–12335 |year=2007 |pmid=17636132 |doi=10.1073/pnas.0702265104 |pmc=1941469|bibcode=2007PNAS..10412330K |doi-access=free }}{{cite journal |vauthors=Adhikari AN, Freed KF, Sosnick TR |title=De novo prediction of protein folding pathways and structure using the principle of sequential stabilization |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=109 |issue=43 |pages=17442–17447 |year=2012 |doi=10.1073/pnas.1209000109 |pmid=23045636 |pmc=3491489|bibcode=2012PNAS..10917442A |doi-access=free }}

• Scoring functions for docking
• Discrete optimized protein energy
• CASP
• CAMEO3D
• Lennard-Jones potential
• Bond order potential

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Category:Potentials

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Category:Molecular modelling

Category:Protein structure