Clifford W Fong
email: cwfong@internode.on.net

Predicting PARP inhibitory activity -A novel quantum mechanical based model

Keywords: PARP, inhibitor binding, desolvation, electron affinity Abbreviations

It has been shown that a novel 4 parameter model (IC 50 =ΔG desolvation free energy of water + lipophilicity + dipole moment + molecular volume) applies to a broad range of PARP1 and PARP2 inhibitory activity. Similarly to other inhibitors, desolvation plays a critical role in binding activity, a factor usually ignored in quantitative binding models. The model is consistent with a process where the water solvated inhibitor moves from a bulk water environment ε 78.3 to an environment representative of the interior of a protein binding pocket ε 6-10. Correlations have been found between equation 1 and literature docking studies which support this model.

A correlation has been found using a variant of equation 1 using the electron affinity of the inhibitors (along with the desolvation, lipophilicity and dipole moment). This observation may support the notion that PARP inhibitors that show synergism with adjuvant drugs and radiotherapy that are used concomitantly with PARP inhibitors in the clinic may involve redox electron transfer processes.

Introduction

The poly(ADP-ribose) polymerase (PARP) superfamily includes at least 17 enzymes involved in several biological processes, including transcriptional regulation, DNA repair, cell cycle regulation, inflammation, hypoxic response, spindle pole function, oncogenerelated signalling, and cell death. PARP1 is the most abundant and best characterized protein within this large family. PARP1 is a DNA repair enzyme but has a role in cell homeostasis at several levels, including transcriptional and epigenetic mechanisms. PARP2 displays almost overlapping functions with PARP1. PARP3 is involved in DNA repair, mitotic spindle integrity, and telomerase regulation. PARP couples with BRCA. Germline inactivation of BRCA1/2 tumor suppressor genes strongly predisposes to breast and ovarian cancer as well as other malignancies, including pancreatic and prostate cancer. The targeting of tumour suppressor defects using PARP inhibitors has been effective in BRCA-related tumours due to sabotage of functionally interdependent DNA repair routes that ultimately leads to cellular toxicity. PARP inhibitors improve progression-free survival in women with recurrent platinum-sensitive ovarian cancer, particularly by olaparib added to conventional treatment. Olaparib was US FDA approved in 2014 as monotherapy (at 400 mg taken twice per day) for patients with germline BRCA mutated (gBRCAm) advanced ovarian cancer who have been treated with three or more prior lines of chemotherapy. Rucaparib was granted breakthrough therapy designation by the FDA in 2015, for use as monotherapy in patients with BRCA1/2 mutant (germline or somatic) advanced ovarian cancer after at least two prior lines of platinum-containing therapy. Several other PARP inhibitors are currently undergoing clinical trials. [START_REF] Brown | PARP inhibitors: the race is on[END_REF][START_REF] Lupo | Inhibition of poly(ADP-ribosyl)ation in cancer: Old and new paradigms revisited[END_REF][START_REF] Mangerich | How to kill tumor cells with inhibitors of poly(ADPribosyl)ation[END_REF][START_REF] Underhill | A review of PARP inhibitors: from bench to bedside[END_REF] (2016) 114, 713-715 DNA is damaged thousands of times during each normal cell cycle, and that damage must be repaired. PARP1 is important for repairing single-strand breaks ('nicks' in the DNA). If such nicks remain unrepaired until DNA is replicated before cell division, then the replication itself can cause double strand breaks to form. Drugs that inhibit PARP1 cause multiple double strand breaks to form, and in tumours with BRCA1, BRCA2 or PALB2 mutations, these double strand breaks cannot be efficiently repaired, leading to the death of the cells. Normal cells that don't replicate their DNA as often as cancer cells, and cells that lack any mutated BRCA1 or BRCA2 still have homologous repair operating, which allows them to survive the inhibition of PARP. PARP inhibitors have an additional mode of action involving localizing PARP proteins at sites of DNA damage, which has relevance to their anti-tumour activity. The trapped PARP protein-DNA complexes are highly toxic to cells because they block DNA replication. The trapped PARP-DNA complexes are more toxic to cells than the unrepaired single-strand DNA breaks that accumulate in the absence of PARP activity, indicating that PARP inhibitors act as PARP poisons. It is suggested that there may be two classes of PARP inhibitors, catalytic inhibitors that act mainly to inhibit PARP enzyme activity and do not trap PARP proteins on DNA, and dual inhibitors that both block PARP enzyme activity and act as PARP poisons. [START_REF] Murai | Trapping of PARP1 and PARP2 by clinical PARP inhibitors[END_REF] The various classes of PARP inhibitors have been reviewed. [START_REF] Steffen | Structural implications for selective targeting of PARPs[END_REF][START_REF] Halder | Stepwise development of structure-activity relationship of diverse PARP-1 inhibitors through comparative and validated in silico modeling techniques and molecular dynamics simulation[END_REF][START_REF] Prokhorov | Combining 3D-QSAR and molecular docking for the virtual screening of PARP inhibitors[END_REF][START_REF] Fatima | Molecular docking and 3D-QSAR studies on inhibitors of DNA damage signaling enzyme human PARP-1[END_REF] The various classes include PARP1 and PARP2 specific: isoquinolinone, indazole, phenanthridine, benzonaphthyridinone. Efforts to discover and optimize new PARP inhibitors have mainly centred on 3D-QSAR studies. 3D-QSAR involve the application of force field calculations requiring three-dimensional structures, e.g. based on protein crystallography or molecule superimposition. It uses computed potentials, e.g. the Lennard-Jones potential, rather than experimental constants and investigates with the overall molecule rather than a single substituent. It examines the steric fields (shape of the molecule), the hydrophobic regions (water-soluble surfaces), and the electrostatic fields. Many such studies have incorporated accompanying computational docking studies. [START_REF] Steffen | Structural implications for selective targeting of PARPs[END_REF] Other studies have used detailed molecular dynamics, principal components analysis and conformational analysis to investigate binding mode of PARP inhibitors. [START_REF] Salmas | In silico investigation of PARP-1 catalytic domains in holo and apo states for the design of high-affinity PARP-1 inhibitors, Inhib[END_REF] A detailed analysis of the x-ray structures of the PARP family interaction with inhibitors has identified the common interactions between ligands and the conserved glycine (backbone) and serine (side-chain hydroxyl) residues and the space occupied by various PARP inhibitors bound to human PARP catalytic domains. [START_REF] Wahlberg | Family-wide chemical profiling and structural analysis of PARP and tankyrase inhibitors[END_REF] It is clear that the hydrogen bonding interactions, via glycine and serine, are important linkages, but the non-polar or hydrophobic interactions between the inhibitors and the enzyme are also key factors in binding interaction. Clearly the molecular size and hydrophobic areas of the inhibitors, besides the hydrogen bonding interactions, play important roles and must be critical factors in predicting the inhibitory capacity of potential inhibitors.

When a small-molecule binds to a protein, it must displace most of the waters occupying the binding cavity, but rarely are all water molecules displaced. A comprehensive analysis of 392 high-resolution crystal structures of proteins with interacting ligands showed that in >84% of the complexes, one or more waters are present and interact with the ligand mediating its interaction with the protein. [START_REF] Lu | Analysis of ligand-bound water molecules in high-resolution crystal structures of protein-ligand complexes[END_REF] Some waters can be so strongly bound and conserved among similar proteins they are considered a part of the target structure, altering the binding site structure. Classical examples are HIV-1 protease where stable bridging waters are targeted to increase inhibitor affinity can play a critical role in HIV-1 resistance. [START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Adachi | Structure of HIV-1 protease in complex with potent inhibitor KNI-272 determined by high-resolution X-ray and neutron crystallography[END_REF] Stable waters can also be displaced to improve ligand affinity by the entropy gain resulting from the release of ordered water to the bulk solvent. These strategies have been successfully applied in designing scytalone dehydratase inhibitors and cyclic urea inhibitors of HIV-1 protease [START_REF] Chen | Structure-based design of potent inhibitors of scytalone dehydratase: displacement of a water molecule from the active site[END_REF][START_REF] Lam | Rational design of potent, bioavailable, nonpeptide cyclic ureas as HIV protease inhibitors[END_REF]. Weakly bound waters are more likely to play varying roles depending upon the nature of the bound ligand. The same water can be stabilized by one ligand and displaced by another, as with poly(ADP-ribose) polymerase inhibitors [START_REF] Bellocchi | Docking studies on PARP-1 inhibitors: insights into the role of a binding pocket water molecule[END_REF][START_REF] Garcia-Sosa | Including tightly-bound water molecules in de novo drug design. Exemplification through the in silico generation of poly(ADP-ribose)polymerase ligands[END_REF]. PARP inhibitors have found clinical application as monotherapies (after failure of standard chemotherapies) or combinatorial regimes often with cytotoxic drugs (topoisomerase I and II inhibitors, temozolomide, gemcitabine, Pt drugs etc). Before PARP inhibitors were found to be effective against BRCA-deficient tumours, they were used as chemosensitizers. There is evidence of synergism between certain PARP inhibitors and adjuvant cytotoxic drugs and radiation therapy [START_REF] Lupo | Inhibition of poly(ADP-ribosyl)ation in cancer: Old and new paradigms revisited[END_REF][START_REF] Mangerich | How to kill tumor cells with inhibitors of poly(ADPribosyl)ation[END_REF][START_REF] Underhill | A review of PARP inhibitors: from bench to bedside[END_REF]. The basis of this synergism may be because radiotherapy damages cells by causing DNA breaks, and PARP inhibitors disrupt DNA repair mechanisms. Since free radicals are involved in radiation induced damage to DNA, we have previously suggested that the electron transfer properties of combinatorial drugs such as Pt with adjuvant drugs can lead to synergism between the cytotoxic drugs. [START_REF] Fong | Molecular mechanisms of cellular transport and resistance of platinum anticancer drugs[END_REF][START_REF] Fong | Platinum based radiochemotherapies: free radical mechanisms and radiotherapy sensitizers[END_REF] This study will use a model (equation 1 or 1a) which has been successfully applied to various transport and drug-protein binding processes. A common model for both processes based on 4 molecular physico-chemical properties of the prospective drugs has been previously applied to passive and facilitated diffusion, and active organic anion transporter drug membrane transport, statins-HMGCoA reductase, some competitive statin-CYP enzyme binding processes, tyrosine kinase inhibitors, HIV-1 protease inhibitors, and cyclin-dependent kinases. Equation 1 also applies to the active competitive transport of these tyrosine kinase inhibitors by the hOCT3, OATP1A2 and OCT1 transporters. [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF][START_REF] Fong | Statins in therapy: Cellular transport, side effects, drug-drug interactions and cytotoxicity -the unrecognized role of lactones[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Fong | Drug discovery model using molecular orbital computations: tyrosine kinase inhibitors[END_REF] The model comprises four main properties: (a) desolvation energy in water (b) lipophilicity or hydrophobicity based on the solvation energy in a hydrophobic solvent such as n-octane or n-octanol; lipophilicity being a measure of how well a drug can interact with lipophilic cell membrane bilayers, and hydrophobicity being a measure of non-polar interaction between a drug and the hydrophobic sectors of a protein (c) dipole moment in water, as a measure of the polar attraction between the drug and its receptor target or cell membrane (d) the molecular volume of the drug in water as a measure of how well the prospective drug fits into the cavity of the target receptor protein or active protein transporter, or how well a drug can diffuse through a cell membrane. The basic model is shown in equation 1.

= desolvation + lipophilicity + !" # $" + # !"%&! ' !&$" Or = desol,CDS + lipophilicity + !" # $" + # !"%&! ' !&$"
ΔG desolvation = ΔG electrostatic + ΔG CDS where CDS is the cavity dispersion solvent structure. The CDS involves non-bulk solvent electrostatic contributions to the free energy of hydration. The SMD solvation model is based on ΔG S o = ΔG ENP + G CDS where ENP is the electronic nuclear polarization: the change in the solute free energy due to electrostatic interactions between the solute and the bulk solvent and distortion of the solute's electronic structure in solution. The solvent is modelled as a dielectric continuum. The CDS represents first solvation shell effects. It involves atomic surface tension (geometry dependent proportionality constants). The CDS has been parameterized using extensive experimental data sets for optimization, and has the advantage of including a realistic experimentally based hydrogen bonding model. The CDS covers shorter-range polarization effects and shorterrange non-electrostatic effects such as cavitation, dispersion, and solvent structural effects (which includes both hydrogen bonding) and exchange repulsion effects. [START_REF] Marenich | Universal Solvation Model Based on Solute Electron Density and on a Continuum Model of the Solvent Defined by the Bulk Dielectric Constant and Atomic Surface Tensions[END_REF] ΔG desol,CDS can be substituted for ΔG desolvation where this non-polar term may be more representative of the interaction processes being studied, as for example in a ligand-protein binding study within the protein binding pocket where the dielectric constant for that environment is ca. ε=6-10 compared to ε=78.3 for the bulk water environment where the use of ΔG desolvation is more appropriate. [START_REF] Fong | Statins in therapy: Cellular transport, side effects, drug-drug interactions and cytotoxicity -the unrecognized role of lactones[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Fong | Drug discovery model using molecular orbital computations: tyrosine kinase inhibitors[END_REF] It has been shown that continuum based hydrophobic non-polar hydration solvation models can fairly accurately model particular protein-protein and ligand-protein complexes, although different protein systems may have unique attractive (van der Waals) properties not precisely transferable to other systems. [START_REF] Levy | On the Nonpolar Hydration Free Energy of Proteins: Surface Area and Continuum Solvent Models for the Solute-Solvent Interaction Energy[END_REF] The hydration hydrophobic effect is primarily due to a decrease in the number of ways that favourable hydrogen bonding can be achieved by solvent water molecules when they are near to a non-hydrogen bonding solute, thereby decreasing the solvent's entropy. The hydrophobic non-polar hydration free energy ΔG non-polar can be considered to comprise of the repulsive ΔG cavity and attractive ΔG vdw terms. The ΔG cavity term includes the repulsive electrostatic effect of forming a cavity in the solvent whereby the solute interacts with the water solvent, while the ΔG vdw term includes shorter-range polarization effects and shorter-range non-electrostatic effects such as cavitation, dispersion, and solvent structural effects (which includes both hydrogen bonding) and exchange repulsion effects.

When a solvated ligand enters a protein binding cavity and begins to bind with the protein, the desolvation processes involving the protein and ligand may involve extensive rearrangement of water molecules, and possibly some expulsion of water molecules from the cavity. A significant change in solvation entropy should then occur on the side on the ligand and the protein receptor.

We have previously shown that the ΔG desolv,CDS values may be a close approximation of how a solvated inhibitor reacts as it approaches and starts to enter the binding pocket (dielectric constant ε=20-30) and leaves the bulk water environment (ε =78.3) and binding starts to occur. [START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Fong | Drug discovery model using molecular orbital computations: tyrosine kinase inhibitors[END_REF] This scenario has been modelled using more intensive molecular mechanics computations. [START_REF] Setny | How Can Hydrophobic Association Be Enthalpy Driven?[END_REF][START_REF] Mondal | Role of Desolvation in Thermodynamics and Kinetics of Ligand Binding to a Kinase[END_REF][START_REF] Shan | How Does a Drug Molecule Find its Target Binding Site?[END_REF] It has also been shown [START_REF] Fong | Statins in therapy: Cellular transport, side effects, drug-drug interactions and cytotoxicity -the unrecognized role of lactones[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Fong | Drug discovery model using molecular orbital computations: tyrosine kinase inhibitors[END_REF] that equation 1 can be modified to model redox processes (eg metabolic oxidation of statins by CYP enzymes) particularly when the molecular volume changes are small. The electron affinity EA or ionization potential IE in water are measures of the ease that drugs can be reduced or oxidised by appropriate species such as enzymes.

," -% . / = desolvation + lipophilicity + !" # $" + 0!"% 122 /

Results

In this study, three sets of PARP inhibitors have been evaluated by applying equation 1 to known experimental IC 50 data. [START_REF] Steffen | Structural implications for selective targeting of PARPs[END_REF][START_REF] Salmas | In silico investigation of PARP-1 catalytic domains in holo and apo states for the design of high-affinity PARP-1 inhibitors, Inhib[END_REF][START_REF] Torrisi | Discovery and SAR of novel, potent and selective hexahydrobenzonaphthyridinone inhibitors of poly(ADPribose)polymerase-1 (PARP-1)[END_REF] The study of the hexahydrobenzonaphthyridinone inhibitors [START_REF] Torrisi | Discovery and SAR of novel, potent and selective hexahydrobenzonaphthyridinone inhibitors of poly(ADPribose)polymerase-1 (PARP-1)[END_REF] has a large and diverse range of inhibitors so the IC 50 experimental data is internally consistent within the experimental error of the experimental conditions. This data is more accurate and internally consistent than the IC 50 data used by the other two studies [START_REF] Steffen | Structural implications for selective targeting of PARPs[END_REF][START_REF] Salmas | In silico investigation of PARP-1 catalytic domains in holo and apo states for the design of high-affinity PARP-1 inhibitors, Inhib[END_REF] where data was collected from multiple studies, where experimental conditions were different, so correlations using these data bases are less robust, and data measurements for the notionally same cell line can vary by ten fold. It should be noted that the equations derived in this study are indicative rather than highly robust, since the experimental data sets are too small to be highly robust or internally cross validated. However, the general model does apply to the three internally consistent data sets which cover a wide range of structurally diverse inhibitors, as well as a number of other biological activities [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF][START_REF] Fong | Statins in therapy: Cellular transport, side effects, drug-drug interactions and cytotoxicity -the unrecognized role of lactones[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Fong | Drug discovery model using molecular orbital computations: tyrosine kinase inhibitors[END_REF]. The model has been validated against the literature docking scores [START_REF] Salmas | In silico investigation of PARP-1 catalytic domains in holo and apo states for the design of high-affinity PARP-1 inhibitors, Inhib[END_REF] for 14 PARP-1 inhibitors (as shown below in eq 5a and 5b).

(a) Analysis of hexahydrobenzonaphthyridinone inhibitors [START_REF] Torrisi | Discovery and SAR of novel, potent and selective hexahydrobenzonaphthyridinone inhibitors of poly(ADPribose)polymerase-1 (PARP-1)[END_REF] The inhibitors analysed are shown in Figure 1. Preliminary analysis showed that inhibitors 4, 7, 20 and 45 were large outliers as neutral species, but fitted the correlation when treated as protonated ions of the amine bases. Protonation of these bases is consistent with similar compounds at the physiological pH (ca 7). ?BK\@G] ^KBND = +9. 9;; desolvation + 9. >=; lipophilicity + 9. 99> ?@ABCD EBFDGH -9. 9>; EBCDKLCMN OBCLFD -2.91

(7) (8) (9) ( (1) (2) (3) (4) (5)
) (11) (12) (13) 10
(4) (5) (6) (7) (8) (9) (1) (2) (3)
Where R 2 = 0.730, SEE = 1.414, SE(ΔG desolvation ) = 0.080, SE(ΔG lipophilicity ) = 0.185, SE(Dipole Moment) = 0.038, SE(Molecular Volume) = 0.008, F=5.33, Significance F = 0.017 Docking Score values for 14 PARP-1inhibitors Eq 5b ?BK\@G] ^KBND = +9. >>T desolvCDS + 9. 9I; lipophilicity -9. 997 ?@ABCD EBFDGH -9. 9>T EBCDKLCMN OBCLFD + 10.1

Where R 2 = 0.692, SEE = 1.44, SE(ΔG desolv,CDS ) = 0.370, SE(ΔG lipophilicity ) = 0.050, SE(Dipole Moment) = 0.040, SE(Molecular Volume) = 0.007, F=5.04, Significance F = 0.020 (c) Analysis of various PARP1 and PARP2 inhibitors [START_REF] Steffen | Structural implications for selective targeting of PARPs[END_REF] See Figure 3 for structures.

Preliminary analysis found that structure isoquinolone3 consistently was a large outlier from all correlations possible due to different experimental conditions for that inhibitor, and was excluded. lnIC 50 values for 8 PARP-1inhibitors Eq 6b Isoquinolone DPQ was a large outlier and excluded possible due to a specific cause not seen in eq 6a.

! 34 = +>. =7 desolv,CDS + 9. 9< lipophilicity -9. 9U ?@ABCD EBFDGH -9. 9U EBCDKLCMN OBCLFD + 10.1

Where R 2 = 0.857, SEE = 1.91, SE(ΔG desolv,CDS ) = 0.54, SE(ΔG lipophilicity ) = 0.13, SE(Dipole Moment) = 0.16, SE(Molecular Volume) = 0.02, F=4.52, Significance F = 0.122 lnIC 50 values for 9 PARP-2 inhibitors Eq 7a ! 34 50 = -9. <7 desolvation ->. J; lipophilicity -9. 7J ?@ABCD EBFDGH -9. >9 EBCDKLCMN OBCLFD + 12.1

Where R 2 = 0.875, SEE = 1.58, SE(ΔG desolvation ) = 0.18, SE(ΔG lipophilicity ) = 0.43, SE(Dipole Moment) = 0.14, SE(Molecular Volume) = 0.02, F=7.03, Significance F = 0.042 lnIC 50 values for 8 PARP-2inhibitors Eq 7b Isoquinolone DPQ was a large outlier and excluded possible due to a specific cause not seen in eq 7a.

! 34 50 = +>. 77 desolv,CDS -9. 9T lipophilicity -9. >; ?@ABCD EBFDGH -9. 9J EBCDKLCMN OBCLFD + 9.1

Where R 2 = 0.935, SEE = 1.13, SE(ΔG desolv,CDS ) = 0.32, SE(ΔG lipophilicity ) = 0.08, SE(Dipole Moment) = 0.10, SE(Molecular Volume) = 0.01, F=10.87, Significance F = 0.039

Discussion

The results indicate equation 1 applies across a broad range of PARP-1 and PARP-2 inhibitors, and can accurately predict inhibitor activity for a diverse range of structures and ionic species. The correlations with ΔG desolvation and ΔG desolv,CDS are consistent with those found for inhibitors for HIV-1 proteases, cyclin-dependent kinases, tyrosine kinases, statins previously described. [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF][START_REF] Fong | Statins in therapy: Cellular transport, side effects, drug-drug interactions and cytotoxicity -the unrecognized role of lactones[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Fong | Drug discovery model using molecular orbital computations: tyrosine kinase inhibitors[END_REF] These observations are consistent with the notions described in Figure 4, showing that ΔG desolvation describes the bulk water environment and ΔG desolv,CDS describes the water environment in the protein binding pocket. The close similarity between eq 4b and 5b, the correlation with the docking scores from Salmas [START_REF] Salmas | In silico investigation of PARP-1 catalytic domains in holo and apo states for the design of high-affinity PARP-1 inhibitors, Inhib[END_REF] suggest this notion is substantially correct. The docking scores involve consideration of the binding environment in the binding pocket. The difference between eq 5a and 5b represents the change from a bulk water environment (where ΔG desolvation is dominated by electrostatic effects, ε 78.3) to a environment more representative of the environment inside the binding pocket ε 6-10. Eq 5b indicates that water desolvation of the inhibitor is the dominant factor determining binding affinity. The correlations in eq 3a and 3b suggest that redox processes involving electron transfer to PARP inhibitors may be significant in binding interactions with the enzyme as well as interacting with adjuvant drugs used concomitantly with PARP inhibitors in the clinic. Previously observed syngerisms between PARP inhibitors and radiation therapy may involve electron transfer processes. In all cases electron attachment occurred at the -C(=O)-N(H)moiety of the naphthyridinone ring (with the exception of the parent inhibitor ion 4 where the localised HOMO was located on the C atom of the aromatic ring adjacent to the -C(=O)group) which is the site where hydrogen bonding between the serine (side chain) and glycine (backbone) residues of the PARP occurs with the inhibitors. [START_REF] Wahlberg | Family-wide chemical profiling and structural analysis of PARP and tankyrase inhibitors[END_REF] A synergism between 5 fluorouracil and oxaliplatin during radiation therapy involving electron transfer has been previously described [START_REF] Fong | Platinum based radiochemotherapies: free radical mechanisms and radiotherapy sensitizers[END_REF]. So any redox activity would be located at the hydrogen bonding site of the PARP inhibitors at the serine and glycine residue interaction region.

It is noted the correlations for the PARP1 and PARP2 inhibitors shown in eq 6a, 6b, 7a, 7b are very similar, consistent with the close similarity in clinical behaviour of these two classes. [START_REF] Lupo | Inhibition of poly(ADP-ribosyl)ation in cancer: Old and new paradigms revisited[END_REF][START_REF] Mangerich | How to kill tumor cells with inhibitors of poly(ADPribosyl)ation[END_REF][START_REF] Underhill | A review of PARP inhibitors: from bench to bedside[END_REF][START_REF] Murai | Trapping of PARP1 and PARP2 by clinical PARP inhibitors[END_REF] The anomalous large outlier behaviour shown by the isoquinolone DPQ in eq 6b and 7b (but not in eq 6a and 7a) appears to be real and may reflect some strong polar interaction by the -O-(CH 2 ) 4 -N(H + )cyclo(CH 2 ) 5 side chain in the binding pocket.

The correlations in eq 1a and 1b appear to show a negative dependency on desolvation in eq 1b, or a positive solvation effect compared to eq 1a, but this apparent anomaly is not statistically significant, since the SE(ΔG desolv,CDS ) = 8.79 far exceeds the coefficient value (-4.55). However the large coefficient for lipophilicity -7.05 in eq 1b is significant, whereas the value -0.57 in eq 1a is not significant.

Conclusions

It has been shown that the 4 parameter general equation 1 applies to a broad range of PARP1 and PARP2 inhibitory activity. Similarly to other inhibitors [START_REF] Fong | Statins in therapy: Cellular transport, side effects, drug-drug interactions and cytotoxicity -the unrecognized role of lactones[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Fong | Drug discovery model using molecular orbital computations: tyrosine kinase inhibitors[END_REF], desolvation plays a PARP binding pocket critical role in binding activity. The model is consistent with a process where the water solvated inhibitor moves from a bulk water environment ε 78.3 to an environment representative of the interior of a protein binding pocket ε 6-10. Correlations have been found between equation 1 and literature docking studies [START_REF] Salmas | In silico investigation of PARP-1 catalytic domains in holo and apo states for the design of high-affinity PARP-1 inhibitors, Inhib[END_REF] which support this model.

A correlation has been found using a variant of equation 1 using the electron affinity of the inhibitors (along with the desolvation, lipophilicity and dipole moment). This observation may support the notion that PARP inhibitors that show synergism with adjuvant drugs and radiotherapy that are used concomitantly with PARP inhibitors in the clinic [START_REF] Underhill | A review of PARP inhibitors: from bench to bedside[END_REF] may involve redox electron transfer processes.

Materials and Methods

All calculations were carried out using the Gaussian 09 package at the B3LYP/6-31 G**(6d, 7f) level of theory with optimised geometries in water, as this level has been shown to give accurate electrostatic atomic charges, and was used to optimize the IEFPCM/SMD solvent model. With the 6-31G* basis set, the SMD model achieves mean unsigned errors of 0.6 -1.0 kcal/mol in the solvation free energies of tested neutrals and mean unsigned errors of 4 kcal/mol on average for ions. [START_REF] Marenich | Universal Solvation Model Based on Solute Electron Density and on a Continuum Model of the Solvent Defined by the Bulk Dielectric Constant and Atomic Surface Tensions[END_REF] The 6-31G** basis set has been used to calculate absolute free energies of solvation and compare these data with experimental results for more than 500 neutral and charged compounds. The calculated values were in good agreement with experimental results across a wide range of compounds. [START_REF] Rayne | Accuracy of computational solvation free energies for neutral and ionic compounds: Dependence on level of theory and solvent model[END_REF][START_REF] Rizzo | Estimation of Absolute Free Energies of Hydration Using Continuum Methods: Accuracy of Partial Charge Models and Optimization of Nonpolar Contributions[END_REF] Adding diffuse functions to the 6-31G* basis set (ie 6-31 + G**) had no significant effect on the solvation energies with a difference of less than 1% observed in solvents, which is within the literature error range for the IEFPCM/SMD solvent model. It is noted that high computational accuracy for each species in different environments is not the focus of this study, but comparative differences between various species is the aim of the study. The use of various literature values for K m , IC 50 to develop the multiple regression equations have much higher uncertainties than the calculated molecular properties. The statistical analyses include the multiple correlation coefficient R 2 , the F test of significance, standards errors for the estimates (SEE) and each of the variables SE(ΔG desolvation ), SE(ΔG lipophilicity ), SE(Dipole Moment), SE (Molecular Volume), SE(Molecular Volume), as calculated from "t" distribution statistics. Residual analysis was used to identify outliers.

Figure 1 Figure 1

 11 Figure 1. Hexahydrobenzonaphthyridinone PARP Inhibitors

( 1 )

 1 ZINC43206370: 2-(4-(Piperidin-3-yl)phenyl) (2) ZINC13860431: 3-(2,4-dioxo-7,8-dihydro (3) ZINC53297239: 2-[2-[4-(1H-indazol-3 (4) ZINC03821234: KU-0058948, or 1-[5-(5) ZINC34031449: 2-(1-propyl-4-piperidinyl) (6) ZINC28467935: 5-hydroxy-N-(3-morpholinopropyl) , SE(ΔG desolvation ) = 0.85, SE(ΔG lipophilicity ) = 1.98, SE(Dipole Moment) = 1..42, Significance F = 0.005 hexahydrobenzonaphthyridinone inhibitors as anion radicals desolv,CDS + 7. >> lipophilicity + 9. T7 ?@ABCD 7I7. <U XCDKHNBG YZZ@G@H[ -332.2 0, SE(ΔG desolv,CDS ) = 10.56, SE(ΔG lipophilicity ) = 2.25, SE(Dipole Moment) = 1. 5.93, Significance F = 0.010 PARP-1 inhibitors [10] See Figure 2 for structures PARP Inhibitors

  )-2H-indazole-7-carboxamide dihydro-5H-thiopyrano[3,4-e]pyrimidin-1-yl)-N-[2-oxo-2-(4-pyrimidin-3-yl)piperazin-1-yl]ethylamino]-3H-quinazolin-4-one -[(3,4-dihydro-4-oxo-1-phthalazinyl)methyl]-2-fluorobenzoyl]hexahydro piperidinyl)-1H-benzimidazole-7-carboxamide morpholinopropyl)-11H-indeno[1,2-c]isoquinoline-9-sulfonamide , SE(Dipole Moment) = 1.31, as anion radicals Eq 3b ?@ABCD EBFDGH , SE(Dipole Moment) = 1.14, See Figure 2 for structures and ref 10 for -2-ylpiperazin-1 fluorobenzoyl]hexahydro-1H-1,4-dia (7) ZINC135362325: 8-fluoro-2,3,4,5-tetrahydro-1H-pyrido[4,3-c]isoquinolin-6-one (8) ZINC00150158: Benzoyleneurea (9) ZINC01590754: 3-Hydroxybenzamide (10) ZINC02041009: INN or 4-Oxo-2-(beta-chloroethyl)-2,3-dihydrobenzo-1,3-ox (11) ZINC08649468: 4-nitrophenanthridin-6-ol (12) ZINC1790819: 4-Hydroxyquinazoline (13) ZINC01512423: 6-keto-5H-phenanthridine-1-carboxylate (14) ZINC00152996: Benzamide lnIC 50 values for 14 PARP-1 inhibitors Eq 4a ! 34 50 = -9. I< desolvation -9. ;> lipophilicity -9. 9U ?@ABCD EBFDGH -9. 9= EBCDKLCMN OBCLFD + 12.2 Where R 2 = 0.926, SEE = 0.98, SE(ΔG desolvation ) = 0.05, SE(ΔG lipophilicity ) = 0.13, SE(Dipole Moment) = 0.02, SE(Molecular Volume) = 0.015, F=41.95, Significance F = 0.0000 lnIC 50 values for 14 PARP-1inhibitors Eq 4b ! 34 50 = +9. =U desolv,CDS + 9. 9< lipophilicity -9. 9J ?@ABCD EBFDGH -9. 9I; EBCDKLCMN OBCLFD + 10.1 Where R 2 = 0.846, SEE = 1.70, SE(ΔG desolv,CDS ) = 0.44, SE(ΔG lipophilicity ) = 0.05, SE(Dipole Moment) = 0.5, SE(Molecular Volume) = 0.008, F=12.22, Significance F = 0.001 Docking Score values for 14 PARP-1 inhibitors Eq 5a

Figure 3

 3 Figure 3. PARP1 and PARP2 inhibitors

Figure 4 .

 4 Figure 4. Schematic representation of ligand-drug leaving the bulk aqueous environment and docking with the PARP enzyme in the binding pocket (ε is dielectric constant)

Table Hexahydrobenzo

 Hexahydrobenzo

	Isoquinolinone 5-	17.75(-5.28)	-14.95	9.55	238
	11 + e -AIQ	59.7(-5.28) 13.9 (-4.76) -8.8 30.8	22.3 8.65	229 99	1.31
	15 Phenanthridine PJ34 66.7 (-8.90) -38.3 25.6 (-4.85) -15.4	10.4 21.2	242 197
	15 + e -Isoquinolinone DPQ 69.6 (-6.79) -40.7 64.55(-4.05) -30.1	22.0 27.9	247 202	1.42
	16	16.07 (-.54)	-14.24	6.6	244
	16 + e -	51.1(-5.54)	-20.3	21.0	204	1.34
	17	16.1 (-5.15)	-14.1	8.8	255
	17 + e -	51.1(-5.15)	-28.4	22.0	244	1.37
	18	16.6 (-5.51)	-14.2	9.73	216
	18 + e -	56.7(-5.51)	-29.15	21.8	214	1.34
	20 Ion	87.7 (-8.19)	-47.8	42.7	302
	20 Ion + e -	61.5(-8.19)	-28.2	37.5	320	1.78
	29	19.9 (-7.84)	-18.8	9.8	293
	29 + e -	60.2(-7.84)	-32.75	31.1	341	1.34
	33	20.8 (-3.98)	-14.6	10.1	186
	33 + e -	60.0(-3.98)	-29.2	21.0	196	1.34
	36	22.3 (-2.93)	-14	7.9	192
	36 + e -	54.2(-2.93)	-15.9	13.6	174	1.34
	39	14.74 (-4.36)	-12.41	9.1	180
	39 + e -	60.2(-4.36)	-30.6	24.2	175	1.31
	45 Ion	78.6 (-4.33)	-42.15	33	237
	45 Ion + e -	64.2(-4.33)	-21.0	50.9	252	1.39
	PARP Inhibitors
	See Figure 2
	(1) ZINC43206370	13.6 (-4.33) -10.1	4.25	136
	(2) ZINC13860431	14.1 (-4.66) -7.2	4.4	83
	(3) ZINC53297239	10.85 (-.14) -9.4	2.55	110
	(4) ZINC03821234	12.6 (-5.35) -9.65	6.4	86
	(5) ZINC34031449	10.05 (-.22) -6.73	5.7	91
	(6) ZINC28467935	16.1 (-6.19) -11.8	2.9	175
	(7) ZINC135362325 8.7 (-7.11) -10.55	10.75	145
	(8) ZINC00150158	31.86 (-.66) -23.9	11.8	328
	(9) ZINC01590754	66.9 (-4.8) -32.25	35.9	306
	(10) ZINC02041009 57.2 (-7.2) -28.9	24.6	311
	(11) ZINC08649468 32.8 (-3.93) -20.1	7	269
	(12) ZINC1790819	67.55 (-.62) -31	41.3	244
	-naphthyridinone PARP Inhibitors (13) ZINC01512423 74.05 (-.78) -31 ΔG desolvation kcal/mol (CDS) ΔG lipophilicity kcal/mol n-Octane (14) ZINC00152996 52.55 (-.78) -22.3 PARP1 and PARP2	Dipole Moment D Water 37.3 28.85	Molecular Volume 199 150 cm 3 /mol	Adiabatic Electron Affinity
	See Figure 1 Inhibitors	Water			Water	eV Water
	4 Parent Ion 4 Ion + e -See Figure 3 Olaparib	79.35(-4.3) 5.30(-4.3) 25.7 (-8.96) -18.9 -38.95 -16.8	16.8 16.8 6.8	116 116 278	1.54
	7 Ion 7 Ion + e -Veliparib Quinoxaline2	66.2 (-6.0) 34.6(-6.0) 77.3 (-5.93) -39.7 -38 -18.2 16.3 (-4.13) -12.1	9.3 22.6 23.6 7.8	185 194 215 174	1.51
	8 8 + e -Isoquinolinone3 Isoquinolinone4	-13.5 16.1 (-7.05) -13.5 15.9 (-5.22) 22.15(-5.22) -29.0 11 (-7.64) -11.9	9 10.2 8.95 8.3	197 200 197 170	1.31
	9 9 + e -Isoquinolinone5 Niraparib	14.45 (-5.3) 60.8(5.43) 11 (-7.64) 76.4 (-7.63) -43.3 -17.7 -31.0 -11.9	9 21 8.3 39	170 174 194 216	1.34


T Doré
email: dore@inapg.inra.fr

J M Meynard

M Sebillotte

The role of grain number, nitrogen nutrition and stem number in limiting pea crop (Pisum sativum) yields under agricultural conditions

Keywords: Pisum sativum, yield determinants, on-farm diagnosis, nitrogen nutrition, stem number

In a small area of the Paris basin in France, 71 pea (Pisum sativum) crops, cv. Solara, were examined to elucidate the determinants of their yield over three years.

The yields ranged from between 3.71 and 7.81 t.ha -1 , and averaged 5. 85, 6.25 and 6.47 t.ha -1 in 1988, 1989 and 1990 respectively. The large differences observed each year among fields were related essentially to grain number, whereas mean grain weight was less variable. Grain number and pod number were highly correlated. A substantial effect of nitrogen nutrition, assessed by a Nitrogen Nutrition Index on grain number, was observed. The role of reproductive stem number on grain number was also examined, showing a threshold at 115 stems.m -² below which, in the conditions examined, large grain numbers per stem did not compensate for low stem numbers. Furthermore we have also shown that branching might not be sufficient to compensate for low plant numbers in some crops when nitrogen nutrition is inadequate, and this also leads to a decrease in grain number.

INTRODUCTION

The variability in yield of dry peas (Pisum sativum L.) is well-known, but no satisfactory explanations have been found. Numerous authors (see for example [START_REF] Pate | Pea[END_REF][START_REF] Pate | Nodulation and nitrogen metabolism[END_REF][START_REF] Dantuma | Potential and actual yields of dried peas in north western Europe[END_REF][START_REF] Davies | Pea (Pisum sativum L.)[END_REF][START_REF] Heath | Agronomic problems associated with the pea crop[END_REF] mentioned the existence of "unfavourable factors" in on-farm conditions, which may prevent the potential yield of a variety from being attained. But, unlike vining-peas [START_REF] Hardwick | Variability in number of pods and yield in commercial crops of vining peas (Pisum sativum L.)[END_REF], no research work has been published that attempts to explain the differences in yield in combining peas. To identify the factors concerning soil, climate and cropping systems which are responsible for differences, it is necessary to understand how yield is determined in farmers crops, i. e. to ascertain which yield components or physiological mechanisms are implicated in yield variations. This agronomic diagnosis of yield development in farmers' crops was first used on wheat crops [START_REF] Sebillotte | Sécheresse et fertilisation azotée du blé d'hiver ; essai d'analyse de situations au champ par l'étude des composantes du rendement[END_REF] and further extended to other crops and locations [START_REF] Sebillotte | Some concepts for analysing farming and cropping systems and for understanding their different effects[END_REF][START_REF] Meynard | Diagnostic sur l'élaboration du rendement des cultures[END_REF]. This method is based on surveys on farmers' plots ; the use of such surveys to identify reasons for failures to obtain heavy yields has already been used successfully by other authors [START_REF] Hardwick | Variability in number of pods and yield in commercial crops of vining peas (Pisum sativum L.)[END_REF][START_REF] Boiffin | Elaboration du rendement et fertilisation azotée du blé en Champagne crayeuse. I.-Protocole et méthode d'étude d'un problème technique régional[END_REF][START_REF] Durrant | A survey of seedling establishment in sugar-beet crops in 1980 and 1981[END_REF][START_REF] Burleigh | Analysis of factors related to wheat yield on farm fields in the Bureruka highlands of Rwanda[END_REF][START_REF] Shafiq | Diagnosing alternatives in conventional crop rotations : sunflowers as an alternative to wheat in the cotton-based cropping systems of Pakistan's Punjab[END_REF][START_REF] Aubry | Diagnostic des facteurs limitants du rendement du blé dur en parcelles d'agriculteurs dans une petite région du semiaride en Tunisie[END_REF][START_REF] Leterme | Analyse intégrée des rendements du blé tendre et de leurs causes de variation dans un réseau de parcelles d'agriculteurs du Thymerais[END_REF]. It is the opposite but complementary approach to methods commonly used, which involve a detailed empirical study of the effect of one factor, such as waterlogging [START_REF] Cannell | Effects of short-term waterlogging on the growth and yield of peas (Pisum sativum)[END_REF] or soil compaction [START_REF] Hebblethwaite | The effects of soil compaction on the emergence, growth and yield of sugar beet and peas[END_REF] on yield and yield components.

MATERIALS AND METHODS

The study was designed to determine the role of both the different yield components and nitrogen nutrition in yield variations. Data were collected from 32, 27 and 12 farmers ' crops grown in 1988, 1989 and 1990, respectively. All the fields were within an area of c. 1000 km², representative of the central part of the Paris basin (France) and were chosen to represent the diversity of local cropping systems. The soils were loamy with 13 to 37% clay, and the main features of the climate over the growing seasons are described in Table 1. There were very few differences in climate between fields in any one year. In each crop on a farmers' field, an area of ca. 500 m², representative of the crop and well away from the headlands, was selected to carry out all the measures and observations [START_REF] Boiffin | Elaboration du rendement et fertilisation azotée du blé en Champagne crayeuse. I.-Protocole et méthode d'étude d'un problème technique régional[END_REF][START_REF] Durrant | A survey of seedling establishment in sugar-beet crops in 1980 and 1981[END_REF]. We did not interfere in any way with the choices in crop management decided by the farmers. However the selected crops were sown with cultivar Solara, an "afila" variety [START_REF] Hedley | Designing "leafless" plants for improving yields of the dried pea crop[END_REF].

For each crop, the following data were recorded on six replicates of c. 0.5 m² (three or four lines x 0.5m) : number of plants, number of stems at the start of flowering (not available for 1990) and number of reproductive stems (stem carrying at least one grain) at harvest, number of pods (not available for 1990), number of grains per m², mean grain weight at harvest, and aerial dry matter at the start of flowering All the values of yield and mean grain weight are given with a water content of 0%.

The results of [START_REF] Crozat | Relationship between dry matter production and N uptake in pea (Pisum sativum cv. Solara) : a tool for the diagnosis of N deficiency[END_REF] were used to assess nitrogen nutrition. Following [START_REF] Greenwood | Decline in percentage N of C 3 and C 4 crops with increasing plant mass[END_REF] and [START_REF] Lemaire | Relationships between plant-N, plant mass and relative growth rate for C3 and C4 crops[END_REF], these authors established a reference curve for the relationship between aerial dry matter (DM, t.ha -1 ) for Pisum sativum, cv. Solara and nitrogen concentration in aerial dry matter Nc, expressed in %DM, using the equation Nc = 4.66DM -0.17 . The percentage of nitrogen in the pea crop aerial dry matter, Nt, was compared to the reference value, Nc, derived from the reference dilution curve, to give the "N nutrition Index" (NNI) defined as : NNI = Nt/Nc According to [START_REF] Lemaire | Analysis of the effect of N nutrition on dry matter yield of a sward by reference to potential yield and optimum N content[END_REF] and [START_REF] Justes | Determination of a critical nitrogen dilution curve for winter wheat crops[END_REF], the lower the NNI, the poorer the nitrogen nutrition. We thus compared the nitrogen nutrition of crops with very different amounts of growth. The NNI was measured at the start of flowering. This time, about half way through the growth cycle, is the point when the rate of root elongation declines [START_REF] Salter | Root growth as a factor in the response of Pisum sativum L. to irrigation[END_REF], and when nodules are almost completely established on the primary [START_REF] Tricot | Establishment and distribution of pea primary root nodules (Pisum sativum L.) as affected by light shading[END_REF] and secondary roots.

RESULTS

Figure 1 gives the distribution of the yields (Y) obtained for the 3 years in the farmers' fields. For 1988 and 1989, we observed rather similar distributions, with a high dispersion around two close mean values (5.85 and 6.25 t.ha -1 respectively). In 1990, when we studied fewer fields, the mean value was slightly higher (6.47 t.ha -1 ), and the shape of the histogram somewhat different ; we still observed large differences between fields. Correlations among the following variables -yield (Y), mean grain weight (MGW), grain number (GN), reproductive stem number (RSN), plant number (PN), and NNI -were analysed for all the crops of the 3 years. Pod number (PdN) was not included in this analysis, since values were missing for 1990.

At this scale (Table 2), the highest correlations were between yield, grain number and NNI at the beginning of flowering. The reproductive stem number was rather highly correlated with the nitrogen nutrition index, but not so highly correlated with grain number. The correlations between mean grain weight and the other variables were always low, as they were between plant number and the other variables.

Figures 2 and3 show the relationships between yield and respectively grain number and mean grain weight for each one of the 3 years. In 1988 and 1989, there was a close linear relationship between the yield and the number of grains (Fig. 2). In 1990 the relationship was not so close (Fig. 2), but the correlation between Y and MGW was higher (Fig. 3), due to one crop. Coefficient of variations of GN (respectively 12.0, 13.0, and 6. 7% for 1988, 1989 and 1990) and MGW (respectively 6.7, 3.8, and 7.6%) were very different, and it was clear that for each year the large differences in yield observed among fields were essentially due to grain number, though this was less marked for 1990. On addition, linear regressions showed that these differences in grain number were correlated to variation in pod number per m² in 1988 and 1989 (R² of linear regressions between GN and PdN were respectively 0.89 and 0.87 in 1988 and 1989, P<0.001). There were few differences in the mean number of grains per pod between fields in each year, and among years.

The crops with poorest nitrogen nutrition always had low GN values (Fig. 4). A significant linear regression was obtained between GN and NNI :

GN = 607 + 1764NNI (P<0.001, R² = 0.51)
The range in NNI depended on the year : most 1990 crops had an NNI very close to 1, which suggested favourable nitrogen nutrition, whereas several 1988 crops had a small NNI ; 1989 crops were more variable. Due to these differences, a year by year analysis does not make it possible to establish annual relationships, such as for the effect of grain number on yield. The effect of nitrogen nutrition on grain number could be due to an effect of nitrogen on grain number per stem, as shown under experimental conditions by [START_REF] Jeuffroy | Rôle de la vitesse de croissance, de la répartition des assimilats et de la nutrition azotée, dans l'élaboration du nombre de graines du pois protéagineux de printemps (Pisum sativum L.)[END_REF], [START_REF] Crozat | Yield performance and seed production pattern of field-grown pea and soybean in relation to N nutrition[END_REF] and [START_REF] Duthion | Nutrition azotée et production du pois[END_REF]. It might also be the result of an effect of nitrogen on stem number.

For a reproductive stem number larger than 115 stems.m -², there was no GN increase when RSN improved, the largest values being about c. 2800-2900 grains.m -² (Fig. 5). But for values of RSN lower than 115 stems.m -², none of the fields attained this maximum level. There were large differences among years in the proportions of crops with less than 115 stems.m -², which were respectively 84, 19, and 8% in 1988, 1989 and 1990. It seemed that the differences in GN observed were due to (i) differences in the number of reproductive stems, and (ii) differences in the number of grains per stem for a given number of stems.

The reproductive stem number at the start of flowering was greatly dependent on (i) the number of plants and (ii) NNI (Fig. 6). Multilinear regression between RSN on one hand and both the number of plants and the NNI on the other hand showed that a large part of inter-crop differences in stem number.m -² was thus explained by these two factors : RSN = -98.8 + 0.997PN + 131NNI (multiple r = 0.67, P≤0.001 for all coefficients)

The number of stems at harvest was fairly well correlated with the stem number at the start of flowering (SNSF) (R² = 0.62 for the crops from 1988 and 1989, P<0.01 ; data was not available for 1990). This indicates that the main differences observed at harvest already existed at this earlier stage. A similar multilinear regression at the start of flowering gave the following results : SNSF = -158.06 + 1.25 PN + 199NNI (multiple r = 0.75, P<0.001 for all coefficients) Less branching was observed for low values of NNI than for high values.

Considering the results shown in Figure 5, the among-crop differences of mean grain number per stem cannot be analysed without reference to stem number. This is why we did not observe directly the effect of nitrogen nutrition on grain number per stem.

DISCUSSION

When this study was undertaken, it was not known which yield components were responsible for the great variability in pea crop yields within a small geographical area. In the case we studied, differences between plots were very large, far larger than the differences observed between years. It confirmed the high variability of pea crop yields, which has been pointed out by numerous authors to be one of the crop's biggest problems [START_REF] Snoad | Improving the pea crop[END_REF][START_REF] Heath | Agronomic problems associated with the pea crop[END_REF]. During the 3 years of this research, and in particular the first 2 years, grain number was far more important than mean grain weight in explaining yield differences.

The limited effect of the end of the pea growth cycle was probably due to favourable growth conditions at late stages, with no constraint from the lack of water in northern areas of France during grain filling, but unfavourable conditions for diseases due to relatively dry weather. These phenomena, widely observed for other grain crops such as wheat or corn in similar areas, may not be as true in drier or in more rainy areas. To improve the stability of pea crop yields requires high grain numbers to be achieved more regularly.

The high correlation which characterised the relationship between grain number and pod number in our results has already been mentioned by other authors [START_REF] Belford | Effects of waterlogging at different stages of development on growth and yield of peas (Pisum sativum L.)[END_REF], but is still striking. Pod number per reproductive node is zero, one or two, whereas the maximum grain number per pod, which characterises each variety [START_REF] Davies | Pea (Pisum sativum L.)[END_REF], is nine for Solara.

According to the hypothesis that all the ovules are fertilised [START_REF] Linck | The morphological development of the fruit of Pisum sativum, var. Alaska[END_REF], and following the results of [START_REF] Jeuffroy | A model to predict seed number per pod from early pod growth rate in pea (Pisum sativum L.)[END_REF] who showed the major effect of assimilate supply on the regulation of grain number per reproductive node, one might suppose that the opportunity for regulation is greater for grain number per pod than for pod number per node. Nevertheless, [START_REF] Linck | The morphological development of the fruit of Pisum sativum, var. Alaska[END_REF] obtained about 35 to 45% of ovules which became grains in pods (that is to say 3.05 to 4.05 grains per pod for Solara), and we obtained about 3.5 to 4.5 grains per pod, which is a very closed value ; and this stability of mean grain number per pod has often been underlined in experimental conditions (for example [START_REF] French | The contribution of pod numbers to field pea (Pisum sativum L.) yields in a short growing-season environment[END_REF].

The correlation we observed between NNI and GN may mean that nitrogen nutrition was in some situations a limiting factor for grain number, as it was a limiting factor for growth when NNI values were lower than one. We cannot be sure of this because the grain formation period takes place mostly after the start of flowering, when we measured NNI. Nevertheless, the causes of low NNI have been identified elsewhere [START_REF] Doré | Analyse, par voie d'enquête, de la variabilité des rendements et des effets précédent du pois protéagineux de printemps (Pisum sativum L.)[END_REF], and it was shown that nitrogen supply probably followed the same pattern for each plot before and after the start of flowering. This effect of nitrogen nutrition on yield determination in pea crops had never been shown before in farmers' conditions in temperate areas, despite the many papers devoted to nitrogen nutrition of peas (see for example the reviews of [START_REF] Pate | Nodulation and nitrogen metabolism[END_REF][START_REF] Sprent | Environmental effects on nitrogen fixation[END_REF][START_REF] Streeter | Inhibition of legume nodule formation and N 2 fixation by nitrate[END_REF][START_REF] Sprent | Effects of environment on temperate grain legume physiology, with an emphasis on nitrogen fixation[END_REF]Doré, 1994a). The result has two important consequences. First, it is important to understand how nitrogen nutrition may become impaired, so that it can be improved where it is responsible for low yields. This is a concern not only for agronomy but also for microbiology, physiology and soil science. [START_REF] Sprent | Effects of environment on temperate grain legume physiology, with an emphasis on nitrogen fixation[END_REF] has shown the effect of the root and nodule environment on nitrogen fixation. In this survey for example, other results (Doré, 1994a) indicated that the major parameter responsible for differences in nitrogen nutrition was nitrogen fixation. These differences were due partly to insect damage by Sitona lineatus L. [START_REF] Doré | On-farm approach of attacks by the pea weevil (Sitona lineatus L.; Col., Curculionidae) and the resulting damage to pea (Pisum sativum L.) crops[END_REF] and to seedbed structure. Other factors might be important elsewhere (see the reviews cited above). Second, there is a need to improving knowledge of the physiological mechanisms involved in these effects of nitrogen nutrition.

Our study showed that branching, which is known to compensate for low plant numbers [START_REF] Hedley | Designing "leafless" plants for improving yields of the dried pea crop[END_REF], is not always sufficient on commercial farms to give the maximum grain number. Under our conditions, for Solara, 115 stems.m -² was a threshold value below which the maximum grain number ca. 2900 grains.m -² could not be obtained. The shape of this relationship explains the poor correlation between GN and RSN. These results are very consistent with those of [START_REF] Jeuffroy | Rôle de la vitesse de croissance, de la répartition des assimilats et de la nutrition azotée, dans l'élaboration du nombre de graines du pois protéagineux de printemps (Pisum sativum L.)[END_REF], who showed, by modelling the formation of grain numbers on a pea stem, that when the number of stems of a crop is low, the growth rate per stem is higher than when there are many stems, and the number of grains per stem is enhanced.

Nevertheless at the lowest stem numbers, this effect was not great enough to prevent a decrease in number of grains per m². The between-year differences in stem number may reflect the differences in climate, sowing date and conditions of crop establishment. But the stem number was not the only yield component responsible for the grain number. This also explains why reproductive stem number was not correlated with grain number in the global analysis : the grain number for two crops with the same stem number may be very different because the grain number per stem may be extremely variable, and a plot with fewer than 115 stems.m -² may have more grains than a plot with a high stem number.

The role of crop establishment in obtaining high yields had already been observed by numerous authors [START_REF] Meadley | The growth of vining peas. The effect of density of planting[END_REF][START_REF] Hedley | Designing "leafless" plants for improving yields of the dried pea crop[END_REF]Ambrose, 1981, Cousin et al., 1985). Our results showed that branching may compensate for small plant number to some extent only. Studies need to be conducted to understand the origin of low plant numbers and simultaneous low branching in farmers' crops. Some references already exist for plant number, such as those dealing with the effects of diseases [START_REF] Powell | Seed quality in grain legumes[END_REF] or climatic conditions [START_REF] Matthews | Investigations into ways of improving the establishment of oilseed rape and combining peas[END_REF] or tillage practices [START_REF] Snobar | Stand establishment in pulse crops[END_REF] on crop establishment. They are far rarer for branching (Doré, 1994b). Besides, the results showed that the better the nitrogen nutrition, the more likely is branching able to compensate for low plant density. This is an indication that the effect of nitrogen nutrition on yield determination starts early in the growth cycle of the pea.

The factors responsible for between-field differences in yield components may vary with geographical area. Nevertheless, the role of grain number, stem number and nitrogen nutrition on pea crop yield determination observed here may have a broad validity in the plains of northern Europe. Our methodology may be of interest in studying the pea crop yield variations in other areas.

Fig. 1 .

 1 Fig. 1. Distribution of grain yields for the survey's crops throughout the three years.

Fig. 3 .

 3 Fig. 3. Relationships between mean grain weight and yield for 1988 ( ) ; 1989 ( ) ; and 1990 ( ).

Fig. 4 .

 4 Fig. 4. Relationships between nitrogen nutrition index and grain number for 1988 ( ) ; 1989 ( ) ; and 1990 ( ).

Fig. 5 .

 5 Fig. 5. Relationships between the reproductive stem number and grain number for 1988 ( ) ; 1989 ( ) ; and 1990 ( ).

Fig. 6 .

 6 Fig. 6. Relationships between plant number and reproductive stem number for the 3 years, according to NNI values : NNI≤1 ( ) ; or NNI >1 ( ).

Table 2 .

 2 General correlations (analysis over 3 years, 71 crops) between yield components and yield (see text for abbreviations)

		Yield	Grain Number	Mean Grain	Plant Number Reproductive
				Weight		Stem Number
	Grain Number	0.881
	Mean Grain	0.296	-0.184
	Weight
	Plant Number	0.223	0.328	-0.243
	Reproductive	0.461	0.468	-0.004	0.543
	Stem Number
	Nitrogen	0.716	0.721	0.017	0.362	0.579
	Nutrition Index

ACKNOWLEDGEMENTS

We wish to thank C Barrier and Y Cariou for technical assistance, the farmers for accepting our presence on their fields, and the Institut National de la Recherche Agronomique and the Union Nationale Interprofessionnelle des plantes riches en Protéines for financial support.