Structure-Activity Relationships on the Molecular Descriptors Family Projects at the End

Lorentz JÄNTSCHI¹ and Sorana D. BOLBOACĂ²

¹Technical University of Cluj-Napoca, Romania

²“Iuliu Haţieganu” University of Medicine and Pharmacy Cluj-Napoca, Romania

lori@academicdirect.org, sorana@j.academicdirect.ro

Abstract

Molecular Descriptors Family (MDF) on the Structure-Activity Relationships (SAR), a promising approach in investigation and quantification of the link between 2D and 3D structural information and the activity, and its potential in the analysis of the biological active compounds is summarized. The approach, attempts to correlate molecular descriptors family generated and calculated on a set of biological active compounds with their observed activity. The estimation as well as prediction abilities of the approach are presented. The obtained MDF SAR models can be used to predict the biological activity of unknown substrates in a series of compounds.

Keywords

Structure-Activity Relationship (SAR); Molecular Descriptors Family (MDF); Model Assessment.

                        Introduction

            Structure-Activity Relationships (SARs), Structure-Property Relationships (SPRs) and Property-Activity Relationships (PARs) arise with the studies of Louis Plack HAMMETT in 1937 [1]. Since then, the Hammett’s equation found a lot of applications [2].

            Quantitative relationships (QSAR, QSPR, QPAR) occurs when the property and/or activity are a quantitative one. Not all properties and activities of chemical compounds can be classified as being quantitative. Two interesting examples are LD₅₀ (Median Lethal Dose, 50%)-dose necessary to kill half of the test population, and Sweetness (one of the five basic tastes, being almost universally related as a pleasure experience) of sugars, which can be appreciated only through comparison (relative scale), and we don’t have two references and a scale (such as are boiling and freezing point and Celsius scale for temperature). Neither unanimous accepted as being quantitatively expressed properties does not have same accuracy degree expressed. From this reason in the last time are avoided to be used QSAR, QSPR, and QPAR, in their place being used (Q)SAR, (Q)SPR, and (Q)PAR, or more simple SAR, SPR, and PAR.

            Moving the attention to the structure of compounds, the things are not so complicated. For example, an atom or a bound can exist and their existence can be identify through electronic transitions and/or molecular vibrations and/or rotations or can not (it is a problem of “yes” or “no”). The things are a little bit complicated relative to the molecular geometry particularly in liquid or gas phases. Heisenberg principle presents the uncertainly rules at micro level (molecular and atomic level) [3]. Note that the molecular geometry depends on the environment on which molecule stays (vicinity of the molecule), temperature, pressure, and so on. From this point of view, dealing with molecular geometry is at least a matter of relativity if it is not a matter of uncertainty.

            Thus, in Structure-Property-Activity Relationships (SPARs) approach we work with certainties (as molecular topology), uncertainties (as molecular geometry), relativities (as biological activities) and evidences (as physico-chemical properties).

            The main goal of the researches was to develop an online system able to construct a family of structure based descriptors (called MDF-Molecular Descriptors Family), taking into consideration both geometrical and topological approaches without discrimination, in order to be used in a SAR procedure strengthened with a natural selection algorithm for obtaining best MDF-SAR (Molecular Descriptors Family (based) Structure Activity Relationship) model for given sets of compounds and given property or activity.

MDF Mathematical Model

A mathematical model composed from seven pieces has been developed. Each piece had a list of possibilities related with the physics approach. Every piece gives a letter in the descriptor’s name:

¸ Linearizing operator (1-st letter) make the link between micro, nano, and macro levels. Example: pH = -log[H+] it’s macro property (measure, effect) measured of micro environment (phenomena, cause), the presence and the number of H+ in a given solution. It takes six values.

¸ Molecular level superposing operator (2-nd letter) superposes fragmental contributions. Its existence is sustained by the variety of molecular property/activity causality, from specificity, regio-selectivity, and selectivity (most of biological activities) to structural formula independent (such as relative mass-same for all molecular formula isomers). It takes nineteen values.

¸ Pair-based fragmentation criteria (3-rd letter) implements different criterions. From first SAR studies of Hammet were observed that some parts of a molecule are more active and give the most of the activity/property of a molecule than others (substituent’s role). It takes four values.

¸ Interaction model (4-th letter) implements different levels of approximation (scalar and vector) for superposing of interaction descriptors at fragment level. Are well known that a series of field-type interactions (such as gravitational and electrostatic) are vectorial treated at low range and scalar treated at distance. It takes six values.

¸ Interaction descriptor (5-th letter) implements a series of interaction descriptors for physical entities (such as force, field, energy, potential), how are given in magnetism, electrostatics, gravity and quantum mechanics. It is a fact that different physical entities have different formulas. It takes twenty-four values.

¸ Atomic property (6-th letter) discriminates atoms one to each other through elemental properties. Every atom has a series of characteristics and/or properties making it similar and/or dissimilar to another. It takes six values.

¸ Distance operator (7-th letter) implements both 2D and 3D approaches (topology and geometry). It takes two values.

MDF Physical Model

Each characteristic of the mathematical model is a piece of the physical model. Table 1 presents all possibilities, associated significance and/or formula of MDF physical model. Constructing of descriptors family consists on calculation of 787968 (2 × 6 × 24 × 6 × 4 × 19 × 6) possibilities. Note that not all of them:

o Have a physical meaning (including here logarithm from a negative number, as example).

o Produce finite numbers (including here division by zero, as example).

Two types of degenerations can be observed in descriptors family: (1) a descriptors has the same values for all compounds from the set, and (2) two descriptors with different formula have the same value for all compounds from the set. When these kinds of descriptors are identified, a bias procedure is applied and the descriptors are discarding from the database. The average number of degenerated descriptors for a set of compounds is about 100000.

Table 1. Parameters values of MDF physical model

Nr	Encoding letter no	Parameter	Values
1	7-th (DO)	Distance operator:	Topological distance, `t` Geometrical distance, `g`
2	6-th (AP)	Atomic property:	Cardinality, `C` Count of directly bounded hydrogen’s, `H` Relative atomic mass, `M` Atomic electronegativity, `E` Group electronegativity, `G` Partial charge, `Q`
3	5-th (DIF)	Descriptor of interaction formula:	Distance, `D` = d Inverted distance, `d` = 1/d First atom's property, `O` = p1 Inverted O, `o` = 1/p1 Product of atomic properties, `P` = p1p2 Inverted P, `p` = 1/p1p2 Squared P, `Q` = √p1p2 Inverted Q, `q` = 1/√p1p2 First atom's Property multiplied by distance, `J` = p1d Inverted J, `j` = 1/p1d Product of atomic properties and distance, `K` = p1p2d Inverted K, `k` = 1/p1p2d Product of distance and squared atomic properties, `L` = d√(p1p2) Inverted L, `l` = 1/d√p1p2 First atom's property potential, `V` = p1/d First atom's property field, `E` = p1/d^2 First atom's property work, `W` = p1^2/d Properties work, `w` = p1p2/d First atom's property force, `F` = p1^2/d^2 Properties force, `f` = p1p2/d^2 First atom's property weak nuclear force, `S` = p1^2/d^3 Properties weak nuclear force, `s` = p1p2/d^3 First atom's property strong nuclear force, `T` = p1^2/d^4 Properties strong nuclear force, `t` = p1p2/d^4
4	4-th (IM)	Interaction model:	SP(AP)= ΣvÎFragmentAP(v); CP(AP)=ΣvÎFragmentAP(v)·DO(v,0)/SP(AP) Rare model and resultant relative to fragment's head, `R` DIF(SP(AP),AP(j),CP(AP)) Rare model and resultant relative to conventional origin, `r` DIF(SP(AP),AP(i),CP(AP)) Medium model and resultant relative to fragment's head, `M` ΣvÎFragmentDIF(AP(v),AP(j),DO(v,j)) Medium model and resultant relative to conventional origin, `m` ΣvÎFragmentDIF(AP(v),AP(j),DO(v,0)) Dense model and resultant relative to fragment's head, `D` ΣvÎFragmentDIF(AP(v),AP(j),DO(v,j))´Versor(v,j) Dense model and resultant relative to conventional origin, `d` ΣvÎFragmentDIF(AP(v),AP(j),DO(v,0)) ´Versor(v,j)
5	3-th (FC)	Fragmentation criteria:	Minimal fragments, `m` Maximal fragments, `M` Szeged distance based fragments, `D` Cluj path based fragments, `P`	{i} {v| dGj(v,i)<∞, Gj= G\{j}} {v|d(v,i)<d(v,j)} {v| dGp(v,i)<∞, Gp=G\p; pÎP(i,j)}
6	2-nd (MOSF)	Molecular overall superposing formula:	Conditional, smallest, `m` Conditional, highest, `M` Conditional, smallest absolute, `n` Conditional, highest absolute, `N` Averaged value, sum, `S` Averaged value, average, `A` Averaged value, S/count(fragments), `a` Aver. value, Avg.(Avg./atom)/count(atoms), `B` Averaged value, S/count(bonds), `b` Geometrical, product, `P` Geometrical, mean, `G` Geometrical, P^1/count(fragments), `g` Geometrical, Geom(Geom/atom)/count(atoms), `F` Geometrical, P^1/count(bonds), `f` Harmonic, sum, `s` Harmonic, mean, `H` Harmonic, s/count(fragments), `h` Harmonic, Harm.(Harm/atom)/count(atoms), `I` Harm., s/count(bonds), `i`	Min(IM(f)| f-fragment, IM(f)<∞) Max(IM(f)| f-fragment, IM(f)<∞) Min(Abs(IM(f))| f-fragment, IM(f)<∞) Max(Abs(IM(f))| f-fragment, IM(f)<∞) Σf|IM(f)<∞IM(f) `S`/Σf|IM(f)<∞1 `S`/Σf1 `A`/ΣvÎMol1 `S`/Σ(u,v)ÎMol1 Πf|IM(f)<∞IM(f) (`P`)^1/Σf|IM(f)<∞^1 `S`^1/Σf^1 `G`/ΣvÎMol1 `S`^1/Σ(u,v)ÎMol^1 1/Σ0≠f|IM(f)<∞1/IM(f) Σf|IM(f)<∞1/`s` `s`/Σf1 `H`/ΣvÎMol1 `H`/Σ(u,v)ÎMol1
7	1-st (LO)	Linearization operator:	Identity (no change), `I` Inversed I, `i` Absolute I, `A` Inversed A, `a` Logarithm of A, `L` Logarithm of I, `l`	f(x)=x f(x)=1/x f(x)=|x| f(x)=1/|x| f(x)=ln(x) f(x)=ln(abs(x))

MDF Methodology

MDF uses the data for a given set of molecules:

¸ Input:

o Molecular and/or structural formulas;

o Property/activity values;

¸ Output:

o MDF of the set.

            Following steps are applied:

¸ Draw (by hand) the topological model (2D) of every molecule from the set using HyperChem;

¸ Build (by software) the geometrical model (3D) of every molecule from the set using HyperChem;

¸ Apply (by software) a semiempirical model (for calculating the partial charge distribution on atoms) and (sometimes) a quantum mechanics model (going till most advanced ones such as Ab-initio and Time-Dependent Density Functional Theory) using specific modules of HyperChem (examples: HyperNewton, HyperGauss, HyperNDO) in order to obtain a optimized geometrical model in vitro or in vivo;

¸ Generate (using MDF Software) the MDF family;

¸ Apply the bias procedure;

¸ Obtain simple linear regression relationships between MDF members and given property/activity.

Multivariate MDF-SARs

Client-server applications for multivariate regressions using MDF members was build using Borland Delphi (v.6) and FreePascal (v.2). The applications use MySQL dynamic libraries to connect to MDF database. Following was subject of implementation:

¸ Systematic search (natural selection) in two independent variables (MDF members acting as independent variables);

¸ Systematic search in three independent variables (one being given by name as input data);

¸ Systematic search in four independent variables (two being given as input data);

¸ Systematic evolutionary search in N (N>2) variables (pair of two are natural selected based on input data from regression analysis in N-2 variables);

¸ Random search in N variables.

            Note that a systematic search in three or more variables (with no input fixed variable) is too time and memory expensive (for three variables takes ~2Gb memory, ~120 days).

MDF-SAR Methodology

            Followings act as input data in MDF-SAR approach:

¸ Topological (2D) and geometrical (3D) model of molecules from the set (HyperChem file);

¸ Values of the property/activity of a given set;

¸ Equation(s) with one or more MDF members;

¸ Estimated/predicted values of given property/activity with other SAR models (from specialty literature).

            Following procedures were developed and used:

¸ Browse or Query MDF-SARs by sets. The application displays the obtained MDF-SARs models (including equation, determination coefficient, number of dependent variables, number of molecules in the set) for a selected set when the Browse mode are choused. When query mode are preferred, measured, estimated, and predicted (leave-one-out procedure) values are displayed, as well as cross-determinations between dependent variables are computed.

¸ Leave-one-out procedure (used as well in Query module) need independent variable values (measured property) and dependent variables values (structural descriptors) as input data for every molecule and produces (running inside Query module or independent) a column of predicted values (excluding one-by-one a molecule from the set, computing regression equation and using the regression equation for obtaining a prediction for the excluded molecule), and correlates the predicted values with measured property (cross-validated leave-one-out score).

¸ Training-versus-Test application has as input same measured and calculated values as leave-one-out procedure, and split the entire set in two sets (training and test) the number of molecules in training set being a user defined option. The split are made randomly. Using the molecules from training set, the SAR model is obtained. The SAR model is applied then on test set. Descriptive and inferential statistics are calculated on both training and test set.

¸ MDF-SAR Predictor is a featured application which allow to the user to select a learning set from the database (which contains a measured property on a molecules set). On the selected learning set, one or more MDF-SAR equations are proposed and the user must chouse just one. Using the selected MDF-SAR equation, the user can submit a molecule in HIN format of which structural model were obtained using same level of approximation.

¸ Steiger’s Z test is used for comparison of two or more linear models, in order to see if one is significantly different from another. The procedure, known as correlated correlations, require the measured values, the estimated values by one model, and the estimated values by the another model, from which three correlation coefficients and sample size acts as input data for calculating Z distribution, from which the probability of identity are calculated.

MDF-SAR on Drug Design

            This facility of MDF-SAR allows that having:

¸ A set of compounds of interest with known values of property/activity and an obtained, validated, and stored into the database MDF-SAR

¸ One of more similar/alike with selected set compound(s)

by made of:

¸ MDF-SAR equation

¸ Building of topological (2D) and geometrical (3D) through the same choices as were build on the selected set

to obtain

¸ Predicted value(s) for the property/activity of the new compounds, even if this (these) compound(s) were not yet synthesized, in order to see if the new structure (virtual compound at this time) comes or not with improvements in desired property/activity.

A summary of twenty-seven best performing models in terms of estimation and prediction are presented bellow. The information is summarized according with the investigated activity and compounds classes. The results are expressed as MDF-SAR equation accompanied by the sample size (n), correlation coefficient (r), associated 95% CI of correlation coefficient (95%CI_r), standard error of estimated (s_est), Fisher parameter (F_est) and its type I error of estimated (in round parentheses), prediction power expressed as cross-validation leave-one-out coefficient (r_loo) and its 95% CI (95%CI_rloo), standard error of predicted (s_pred), Fisher parameter (F_pred) and its type I error of predicted (in round parentheses). The Ŷ is the estimated activity by the MDF model, and iMDRoQg is for example the name of the molecular descriptors used by the model

1. Hydrophobic vs. hydrophilic character [4] of standard amino acids

Ŷ =-0.58 + 8.5·iMDRoQg

[5]

n = 15, r [95% CI_r] = 0.9514 [0.8565-0.9840], s_est = 0.44, F_est (p) = 124 (5.05·10^-8),

r_loo [95%CI_rloo] = 0.9351 [0.8028-0.9796], s_pred = 0.51, F_pred (p) = 90 (3.26·10^-7).

2. Hydrophobic vs. hydrophilic character [6] of standard amino acids

Ŷ = 12-21·IGDROQg

[5]

n = 15, r [95% CIr] = 0.9759 [0.9270-0.9921], s_est = 0.71, F_est (p) = 260 (5.66·10^-10),

r_loo [95%CIrloo] = 0.9659 [0.8929-0.9894], s_pred = 0.80, F_pred (p) = 203 (2.57·10^-9).

3. Hydrophobic vs. hydrophilic character [7] of standard amino acids

Ŷ = 81.72 + 817.95·inMrpQg

[8]

n = 20, r [95% CI_r] = 0.9232 [0.8126-0.9695], s_est = 20.73, F_est (p) = 104 (6.69·10^-9),

r_loo [95%CI_rloo] = 0.9082 [0.7727-0.9645], s_pred = 22.58, F_pred (p) = 85 (3.16·10^-8).

4. Hydrophobic vs. hydrophilic character [9] of standard amino acids

Ŷ = 1.36-0.20·iIPmLQt

[8]

n = 20, r [95% CI_r] = 0.9252 [0.8172-0.9704], s_est = 0.36, F_est (p) = 107 (5.30·10^-9),

r_loo [95%CI_rloo] = 0.9003 [0.7546-0.9613], s_pred = 0.42, F_pred (p) = 75 (8.02·10^-8).

5. Hydrophobic vs. hydrophilic character [6] of standard amino acids

Ŷ = -7.60 + 19.17·IiDRLQt

[8]

n = 20, r [95% CI_r] = 0.9328 [0.8348-0.9734], s_est = 1.11, F_est (p) = 120 (2.10·10^-9),

r_loo [95%CI_rloo] = 0.9226 [0.8062-0.9702], s_pred = 1.18, F_pred (p) = 103 (7.25·10^-9).

6. Hydrophobic vs. hydrophilic character [10] of standard amino acids

Ŷ = 0.86-0.96·lAmrLQg

[8]

n = 20, r [95% CI_r] = 0.9376 [0.8461-0.9754], s_est = 0.12, F_est (p) = 131 (1.09·10^-9),

r_loo [95%CI_rloo] = 0.9263 [0.8149-0.9716], s_pred = 0.13, F_pred (p) = 109 (4.73·10^-9).

7. Hydrophobic vs. hydrophilic character [11] of standard amino acids

Ŷ = 86.05 + 843.88·inMrpQg

[8]

n = 19, r [95%CI_r] = 0.9504 [0.8794-0.9805], s_est = 16.49, F_est (p) = 159 (4.77·10^-10),

r_loo [95%CI_rloo] = 0.9380 [0.8428-0.9762], s_pred = 18.37, F_pred (p) = 125 (3.00·10^-9).

8. Water activated carbon adsorption [12] of organic compounds

Ŷ = 2.58 + 0.85·IiMMWHt +0.003·lPMDVQg

[13]

n = 16, r [95%CI_r] = 0.9905 [0.9755-0.9963], s_est = 0.05, F_est (p) = 337 (6.30·10^-12),

r_loo [95%CI_rloo] = 0.9873 [0.9654-0.9953], s_pred = 0.06, F_pred (p) = 251 (4.14·10^-11).

9. Toxicity of Polychlorinated Organic Compounds [14]

Ŷ = 4.06-4.95·imDrkQt + 0.09·LHDROQg+0.06·iSPRtQg

n = 31, r [95% CI_r] = 0.9692 [0.9364-0.9851], s_est = 0.15, F_est (p) = 140 (1.11·10^-16),

r_loo [95%CI_rloo] = 0.9613 [0.9194-0.9816], s_pred = 0.16, F_pred (p) = 109 (3.22·10^-15).

10. Toxicity of mono-substituted nitrobenzene [15]

Ŷ = 6.27-91.15·IBMrkGg

n = 39, r [95% CI_r] = 0.7717 [0.6029-0.8742], s_est = 0.35, F_est (p) = 54 (8.87·10^-9),

r_loo [95%CI_rloo] = 0.7474 [0.5619-0.8612], s_pred = 0.37, F_pred (p) = 48 (4.71·10^-8).

11. Toxicity of benzene derivates [16]

Ŷ = 3.25-9.66·ABmrsQg + 1.00·iGPrfHt

n = 69, r [95% CI_r] = 0.9331 [0.8937-0.9581], s_est = 0.28, F_est (p) = 222 (1.48·10^-30),

r_loo [95%CI_rloo] = 0.9267 [0.8834-0.9542], s_pred = 0.29, F_pred (p) = 201 (2.97·10^-29).

12. Toxicity of alkyl metal compounds [17]

Ŷ = 2.80 + 28.06·IbMmpMg + 0.08·LPPROQg

[18]

n = 10, r [95%CI_r] = 0.9988 [0.9947-0.9997], s_est = 0.06, F_est (p) = 1473 (6.49·10^-10),

r_loo [95%CI_rloo] = 0.9980 [0.9901-0.9995], s_pred = 0.07, F_pred (p) = 841 (4.57·10^-9).

13. Toxicity of para-substituted phenols [19]

Ŷ = 0.09 + 5.56·10^-3·isDDkGg-0.42·IMmrKQg + 9.41·10^-3·lPMDKQg-0.08·lFMMKQg

[20]

n = 30, r [95% CI_r] = 0.9890 [0.9767-0.9948], s_est = 0.17, F_est (p) = 279 (1.10·10^-22),

r_loo [95%CI_rloo] = 0.9839 [0.9655-0.9924], s_pred = 0.21, F_pred (p) = 189 (2.58·10^-20).

14. Relative toxicity of para-substituted phenols [19]

Ŷ = -3.29 + 0.04·ASMmVQt-0.33·lfDdOQg + 0.08·InMrLQg-0.35·LsDMpQg

[21]

n = 30, r [95% CI_r] = 0.9868 [0.9721-0.9937], s_est = 0.12, F_est (p) = 1.50·10^-21,

r_loo [95%CI_rloo] = 0.9823 [0.9621-0.9917], s_pred = 0.14, F_pred (p) = 9.34 10^-20.

15. Cytotoxicity of quinoline [22]

Ŷ = -4.49 + 8.35·INDRLQt + 1.96·lHPmTMt

[23]

n = 15, r [95% CI_r] = 0.9882 [0.9638-0.9961], s_est = 0.17, F_est (p) = 250 (1.65·10^-10),

r_loo [95%CI_rloo] = 0.9805 [0.9377-0.9939], s_pred = 0.22, F_pred (p) = 149 (3.34·10^-9).

16. Mutagenicity of quinoline [22]

Ŷ = -1.57 + 0.21·lNMrSQg + 0.09·ASPrVQg

[23]

n = 14, r [95% CI_r] = 0.9782 [0.9306-0.9932], s_est = 0.18, F_est (p) = 122 (3.12·10^-8),

r_loo [95%CI_rloo] = 0.9666 [0.8891-0.9902], s_pred = 0.22, F_pred (p) = 78 (3.18·10^-7).

17. Antioxidant efficacy of 3-indolyl derivates [24]

Ŷ = 7.18-1.10·lbPMkHg-33.24·iAPrVGt

[25]

n = 8, r [95%CI_r] = 0.9999 [0.9994-0.9999], s_est = 0.01, F_est (p) = 12591 (5.55·10^-10),

r_loo [95%CI_rloo] = 0.9997 [0.9978-0.9999], s_pred = 0.02, F_pred (p) = 3877 (1.05·10^-8).

18. Antiallergic activity of substituted N 4-methoxyphenyl benzamides [26]

Ŷ = -0.15+9.02·10^-4·imMRkMg-0.32·imMDVQg-5.24·10^-5·isDRtHg + 0.14·iHMMtHg

n = 23, r [95%CI_r] = 0.9986 [0.9966-0.9994], s_est = 0.07, F_est (p) = 1638 (7.04·10^-27),

r_loo [95%CI_rloo] = 0.9978 [0.9945-0.9991], s_pred = 0.08, F_pred (p) = 1007 (1.45·10^-24).

19. Antituberculotic activity of polyhydroxyxanthones [27]

Ŷ =-19.11 + 2.32·lHPDOQg +19.34·IsMRKGg

[28]

n = 10, r [95%CI_r] = 0.9987 [0.9942-0.9997], s_est = 0.03, F_est (p) = 1327 (9.33·10^-10),

r_loo [95%CI_rloo] = 0.9974 [0.9871-0.9994], s_pred = 0.04, F_pred (p) = 663 (1.05·10^-8).

20. Growth inhibition activity of taxoids [29]

Ŷ = -17.7 + 0.002·isMdTHg + 77.22·IiDrQHg

[30]

n = 34, r [95% CI_r] = 0.9583 [0.9174-0.9791], s_est = 0.36, F_est (p) = 174 (2.86·10^-18),

r_loo [95%CI_rloo] = 0.9507 [0.9016-0.9755], s_pred = 0.39, F_pred (p) = 146 (2.22·10^-16).

21. Anti-HIV-1 potencies of HEPTA and TIBO derivatives [31]

Ŷ = 17.72-7.11·InMdTHg-1.23·lFDMwEt + 8.36·AiMrKQt + 6.59·10^5·ImDMtQt -5.98·lIMdEMg

[32]

n = 57, r [95% CI_r] = 0.9579 [0.9292-0.9750], s_est = 0.45, F_est (p) = 113 (5.17·10^-28),

r_loo [95%CI_rloo] = 0.9485 [0.9133-0.9696], s_pred = 0.49, F_pred (p) = 91 (1.16·10^-25).

22. Inhibition activity on carbonic anhydrase I of substituted 1,3,4-thiadiazole- and 1,3,4-thiadiazoline-disulfonamides [33]

Ŷ = 1.14 + 8.79·10^-2·inPRlQg + 3.52·10^-3·lPDMoMg + 2.43·iAMRqQg + 1.04·inMRkQt

[34]

n = 40, r [95% CI_r] = 0.9579 [0.9212-0.9776], s_est = 0.16, F_est (p) = 97 (9.45·10^-20),

r_loo [95%CI_rloo] = 0.9440 [0.8950-0.9704], s_pred = 0.19, F_pred (p) = 71 (2.22·10^-16).

23. Inhibition activity on carbonic anhydrase II of substituted 1,3,4-thiadiazole- and 1,3,4-thiadiazoline-disulfonamides [33]

Ŷ = -9.99 + 4.56·imDdSCg+2.94·10-3·isDrqQg + 5.20·IIMDQQg + 1.48·lmMrsGg

[35]

n = 40, r [95% CI_r] = 0.9506 [0.9079-0.9737], s_est = 0.17, F_est (p) = 82 (1.85·10^-18),

r_loo [95%CI_rloo] = 0.9383 [0.8846-0.9674], s_pred = 0.19, F_pred (p) = 64 (1.22·10^-15).

24. Inhibition activity on carbonic anhydrase IV of substituted 1,3,4-thiadiazole- and 1,3,4-thiadiazoline-disulfonamides [33]

Ŷ = 0.62 + 0.10·inPRlQg + 9.92·10^-9·iHMMTQt-9.25·IHMDTQg + 1.73·InPdJQg

[36]

n = 40, r [95% CI_r] = 0.9593 [0.9238-0.9784], s_est = 0.16, F_est (p) = 101 (5.03·10^-20),

r_loo [95%CI_rloo] = 0.9505 [0.9069-0.9739], s_pred = 0.18, F_pred (p) = 82 (2.10·10^-18).

25. Inhibition activity of dipeptides [37]

Ŷ = -7.20 + 0.24·IbMmjHg + 0.02·IbPdPHg-0.24·IBMRQCg + 2.08·ImDmEEt-0.04·ImDrFEt

n = 58, r [95% CI_r] = 0.9618 [0.9360-0.9772], s_est = 0.29, F_est (p) = 128 (9.89·10^-30),

r_loo [95%CI_rloo] = 0.9539 [0.9226-0.9726], s_pred = 0.31, F_pred (p) = 145 (1.87·10^-27).

26. Inhibition activity of 2,4-Diamino-5-(substituted-benzyl)-Pyrimidines [38]

Ŷ = 3.78 + 1.62·iImrKHt + 2.37·liMDWHg + 6.40·IsDrJQt-0.09·LSPmEQg

n = 67, r [95% CI_r] = 0.9517 [0.9223-0.9701], s_est = 0.19, F_est (p) = 149 (2.78·10^-32),

r_loo [95%CI_rloo] = 0.9451 [0.9115-0.9661], s_pred = 0.20, F_pred (p) = 130 (1.70·10^-30).

27. Inhibition activity of peptide analogues [39]

Ŷ = 0.81- 5.21·10^-2·lmDRsQg + 1.84·10^-3·iAPrtQg + 240.89·IHMdpMg -9.64·10^-2·IHMdOMg

n = 47, r [95% CI_r] = 0.9697 [0.9459-0.9830], s_est = 0.16, F_est (p) = 165 (1.12·10^-26),

r_loo [95%CI_rloo] = 0.9611 [0.9303-0.9784], s_pred = 0.18, F_pred (p) = 127 (3.06·10^-24).

Conclusions and Final Remarks

            Realized MDF method and their application MDF-SAR proved to be a very good tool for design of chemical compounds. A series of papers given on results section (over fifty) exposed their ability on investigated sets. The idea about realizing of MDF feigned close to finalizing of PhD studies of first author (Prof. Dr. Mircea V. DIUDEA being his PhD Advisor), but method were implemented just in 2004 (see [40], methodology being revised in 2005 [41]). Further studies will be done in this field, another project being started in 2007, having as main objective creating of a procedure for automatic generating of virtual compounds, based on concepts of combinatorial chemistry. A lesson learned: MDF and MDF-SAR shown miscarries of current methods of constructing/optimizing of molecular geometry (being not capable to provide verifiable and reproducible solutions at a reasonable confidence level). Because MDF give too many weight on geometry, a new method will replace the MDF, a method called MDFV (being already online), a much conservative method regarding molecular topology relative to MDF. An online application compute statistics on physical models of best obtained MDF-SARs, being available at: http://l.academicdirect.org/Chemistry/SARs/MDF_SARs/stats/.

            Statistics are:

¸ Contribution of descriptors by sets for best models;

¸ Inclusion of descriptors by sets for best models;

¸ Classification of interactions by sets for best models;

¸ Contribution of descriptors by sets for all models;

¸ Inclusion of descriptors by sets for all models;

¸ Classification of interactions by sets for all models.

            At the end, the best performing model obtained with MDF-SAR [42] as well as the developed methodology for assessing of structure-activity relationships [43] required to be mentioned here.

            As further plans, the study [44] opens a new path in structure-activity relationships approach and will be further investigated.

Acknowledgements

Special acknowledgments from first author to Prof. Mircea V. DIUDEA, his PhD Advisor from 1997 to 2000. Knowledge basis in the field were obtained during this period.

            The MDF project was granted from 2005 to 2007 (ET36). The MDF-SAR part of MDF is granted from 2006 to 2008 (ET108). First author (as principal investigator) and second author (as co-investigator) are gratefully to UEFISCSU Romania for this.

                        References

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