Quantitative structure-activity relationship

# Overview
Quantitative structure-activity relationship (QSAR) is the process by which chemical structure is quantitatively correlated with a well defined process, such as biological activity or chemical reactivity.
For example, biological activity can be expressed quantitatively as in the concentration of a substance required to give a certain biological response.  Additionally, when physiochemical properties or structures are expressed by numbers, one can form a mathematical relationship, or quantitative structure-activity relationship, between the two.  The mathematical expression can then be used to predict the biological response of other chemical structures.
QSAR's most general mathematical form is:

# SAR and SAR paradox
The basic assumption for all molecule based hypotheses is that similar molecules have similar activities. This principle is also called Structure-Activity Relationship (SAR). The underlying problem is therefore how to define a small difference on a molecular level, since each kind of activity, e.g. reaction ability, biotransformation ability, solubility, target activity, and so on, might depend on another difference. A good example was given in the bioisosterism review of Patanie/LaVoie.
In general, one is more interested in finding strong trends. Created hypotheses usually rely on a finite number of chemical data. Thus, the induction principle should be respected to avoid overfitted hypotheses and deriving overfitted and useless interpretations on structural/molecular data.
The SAR paradox refers to the fact that it is not the case that all similar molecules have similar activities.

# Applications

## Chemical
One of the first historical QSAR applications was to predict boiling points.
It is well known for instance that within a particular family of chemical compounds, especially of organic chemistry, that there are strong correlations between structure and observed properties. A simple example is the relationship between the number of carbons in alkanes and their boiling points. There is a clear trend in the increase of boiling point with an increase in the number carbons and this serves as a means for predicting the boiling points of higher alkanes.
A still very interesting application is the Hammett equation, Taft equation and pKa prediction methods.

## Biological
The biological activity of molecules is usually measured in assays to establish the level of inhibition of particular signal transduction or metabolic pathways. Chemicals can also be biologically active by being toxic.  Drug discovery often involves the use of QSAR to identify chemical structures that could have good inhibitory effects on specific targets and have low toxicity (non-specific activity).  Of special interest is the prediction of partition coefficient log P, which is an important measure used in identifying "drug-likeness" according to Lipinski's Rule of Five.
While many quantitative structure activity relationship analyses involve the interactions of a family of molecules with an enzyme or receptor binding site, QSAR can also be used to study the interactions between the structural domains of proteins. Protein-protein interactions can be quantitatively analyzed for structural variations resulted from site-directed mutagenesis..
It is part of the machine learning method to reduce the risk for a SAR paradox, especially taking into account that only a finite amount of data is available (see also MVUE). In general all QSAR problems can be divided into a coding and learning part.

## Data mining
For the coding usually a relatively large number of features or molecular descriptors is calculated, which can lack structural interpretation ability. In combination with the later applied learning method or as preprocessing step occurs a feature selection problem.
A typical data mining based prediction uses e.g. support vector machines, decision trees, neural networks for inducing a predictive learning model.

### 3D-QSAR
3D-QSAR refers to the application of force field calculations requiring three-dimensional structures, e.g. based on protein crystallography or molecule superposition. It uses computed potentials, e.g. the Lennard-Jones potential, rather than experimental constants and is concerned with the overall molecule rather than a single substituent. It examines the steric fields (shape of the molecule) and the electrostatic fields based on the applied energy function.
The created data space is then usually reduced by a following feature extraction (see also dimensionality reduction). The following learning method can be any of the already mentioned machine learning methods, e.g. support vector machines.
In the literature it can be often found that chemists have a preference for partial least squares (PLS) methods, since  it applies the feature extraction and induction in one step.

## Molecule mining
Molecule mining approaches, a special case of structured data mining approaches, apply a similarity matrix based prediction or an automatic fragmentation scheme into molecular substructures. Furthermore there exist also approaches using maximum common subgraph searches or graph kernels.

### Fragment based (group contribution)
It has been shown that the logP of compound can be determined by the sum of its fragments. Fragmentary logP values have been determined statistically. This method gives mixed results and is generally not trusted to have accuracy of more than +/- 0.1 units.

# Applicability Domain
As the use of (Q)SAR models for chemical risk management increases steadily and is also used for regulatory purposes (in the EU: Registration, Evaluation, Authorisation and Restriction of Chemicals), it is of crucial importance be able to assess the reliability of predictions. The chemical descriptor space spanned by a particular training set of chemicals is called Applicability Domain. It offers the opportunity to assess whether a compound can be reliably predicted.