Monika Dash

Saverio Moroni

Anthony Scemama

Claudia Filippi

Perturbatively selected configuration-interaction wave functions for efficient geometry optimization in quantum Monte Carlo

We investigate the performance of a class of compact and systematically improvable Jastrow-Slater wave functions for the efficient and accurate computation of structural properties, where the determinantal component is expanded with a perturbatively selected configuration interaction scheme (CIPSI). We concurrently optimize the molecular ground-state geometry and full wave function -Jastrow factor, orbitals, and configuration interaction coefficients-in variational Monte Carlo (VMC) for the prototypical case of 1,3-trans-butadiene, a small yet theoretically challenging π-conjugated system. We find that the CIPSI selection outperforms the conventional scheme of correlating orbitals within active spaces chosen by chemical intuition: it gives significantly better variational and diffusion Monte Carlo energies for all but the smallest expansions, and much smoother convergence of the geometry with the number of determinants. In particular, the optimal bond lengths and bond-length alternation of butadiene are converged to better than one m Å with just a few thousand determinants, to values very close to the corresponding CCSD(T) results. The combination of CIPSI expansion and VMC optimization represents an affordable tool for the determination of accurate ground-state geometries in quantum Monte Carlo.

I. INTRODUCTION

Quantum Monte Carlo methods are a class of ab initio approaches which solve the interacting Schrödinger equation stochastically. The most widely used variants of QMC are the variational (VMC) and diffusion Monte Carlo (DMC). Thanks to their favorable scaling with the number of particles and the ease of parallelization, they have often been employed to benchmark electronic properties, in particular, total energies of relatively large molecules as well as solids. Recently, it has been shown 1,2 that it is possible to compute derivatives of the energy at the same computational cost per Monte Carlo step as evaluating the energy itself, also when employing large determinantal expansions in the commonly used Jastrow-Slater QMC wave functions. Consequently, one can compute all derivatives necessary for the optimization of the structure of a system very efficiently and, simultaneously, of the variational parameters in the wave function, as was demonstrated on the structural optimization of short polyenes with expansions comprising over 200,000 determinants 2 . These developments also allow us to thoroughly explore the sensitivity of QMC calculations to the choice of the Slater expansion, namely, the set of orbitals which one must correlate and the truncation of the active space. It was shown that an instructed guess of the orbitals based on chemical intuition can lead to significant variations in VMC energies and structures of a molecule as small as butadiene even when employing large expansions in the presence of a Jastrow correlation factor 2 .

To overcome the limitations of an a priori approach in the a) Electronic mail: moroni@democritos.it b) Electronic mail: scemama@irsamc.ups-tlse.fr c) Electronic mail: c.filippi@utwente.nl choice of the orbital set and achieve a compact description of the determinantal component in QMC, a promising alternative is to employ a selected CI algorithm such as the CIPSI (configuration interaction using a perturbative selection done iteratively) method. CIPSI was originally introduced by Huron et al. in 1973 3 and continually tested [4][5][6][7][8][9][10] , improved [10][11][12][13] , and assessed in comparison to full-CI (FCI) expansions 10,14,15 in combination with a variety of orbital descriptions 16 . In the last few years, there has been renewed interest in the development of selected CI approaches [17][18][19][20][21] to accurately calculate the ground-and excited-state energies of small molecules, establishing the competence of these approaches for benchmarking applications. The use of these expansions has however only been marginally explored within the QMC framework [22][23][24][25][26][27] , where it was shown to yield very good DMC energies at the price of employing large CIPSI wave functions.

Here, we complement the perturbatively selected CIPSI determinants with a Jastrow factor and VMC optimization of the full wave function (Jastrow parameters, orbitals and CI coefficients), and investigate the ability of the resultant Jastrow-CIPSI wave functions to obtain accurate molecular geometries in VMC, as well as corresponding VMC and DMC energies, with relatively compact expansions. We focus on the butadiene molecule, where an accurate estimation of the bond length alternation (BLA) is theoretically quite challenging, demanding a proper description of correlation effects, both static and dynamic. As mentioned above, significant variations in energies and structural parameters were observed in VMC when correlating different sets of σ and π orbitals in a fully optimized Jastrow-Slater wave function 2 : with a large expansion comprising over 45,000 determinants in a restricted active space, good agreement with coupled cluster with singles, doubles, and perturbative triples in the complete basis set limit (CCSD(T)/CBS) was obtained in the bond lengths. This number of determinants is however surprisingly large for the description of the ground state of such a small system. Indeed, we find here that a determinantal description with CIPSI yields a much smoother convergence in the structural parameters and QMC energies when the size of the expansion is systematically incremented. Correspondingly, very accurate values for these physical properties are already obtained with only a few thousand determinants.

To construct the CIPSI component of the QMC trial wave functions, we follow two different schemes: one where we systematically "expand" the wave function by adding important determinants at every step, and the other where we first generate an extremely large wave function and subsequently "truncate" it to obtain requisite sizes. For a fixed size of the CI expansion, the "truncation" scheme is expected to be more accurate than the "expansion" one since the overlap of the wave function with the FCI wave function is larger. The difference can be important in systems where the CI coefficients change dramatically with the number of determinants in the wave function 28 , but generation of the large wave function to initiate "truncation" might not be entirely feasible for larger systems where the "expansion" scheme would therefore represent a computationally less expensive route. For small systems such as butadiene, however, we can employ both strategies and, consequently, draw an assessment of their relative performance. Here, we find that the truncation scheme provides a faster and smoother convergence of the bond lengths and VMC energies of butadiene. We note that we are looking at extremely small variations in the structural parameters (well below m Å) when comparing and establishing the convergence. Finally, while we primarily employ multi-configurational self consistent field (MCSCF) canonical orbitals in the CIPSI algorithm and as starting orbitals in QMC, we additionally test the use of natural orbitals. Rather peculiarly, expansions with natural orbitals commensurate in size with those with canonical orbitals consistently converge to slightly higher VMC energies.

The paper is organized as follows. In section II, we describe the CIPSI algorithm and the functional form of the Jastrow-Slater wave functions and, in Section III, we report the computational details. The numerical results obtained for the structural optimization of butadiene and a comparison with our previous Jastrow-CAS results 2 are given in Section IV.

II. METHODS

CIPSI is an iterative CI selection algorithm that allows us to perturbatively select determinants from the FCI space. Starting with an initial reference wave function, additional determinants are added to the expansion based on their effective second-order energy contribution which is required to be greater than a fixed or an iteratively modifiable threshold. Selection iterations can be performed until a target number of determinants is reached or until some other selection criterion is met 29 . In the process, the quality of the wave function is systematically improved and converges towards the FCI solution.

A step-by-step description of the CIPSI algorithm is pro-vided in Refs. 22, 24, 25, and 30. The scheme is briefly iterated here. The reference wave function is composed of a linear combination of Slater determinants, D i , spanning a space, S:

ψ CIPSI = Di∈S c i D i . (1)
Typically, this reference wave function is initially the single determinant with the lowest energy, which is also the case here. At every iteration, the many-body Hamiltonian, Ĥ, is diagonalized in the reference space S to obtain the variationally minimized energy E ref and the corresponding c i coefficients. Then, all the determinants D j outside of this reference space that are connected to S by Ĥ are generated, and their individual contributions to the energy are estimated with the Epstein-Nesbet [START_REF] Epstein | [END_REF][START_REF] Nesbet | Proc. R. Soc. Lond. A[END_REF] perturbation theory,

δE (2) j = | D j | Ĥ |ψ CIPSI | 2 E ref -D j | Ĥ |D j . (2)
If |δE

(2) j | is greater than a given threshold, determinant D j is selected for the next iteration. Summing all these contributions gives E PT2 , namely, the second-order perturbative energy correction to E ref , and the CIPSI energy of the current iteration is given by

E CIPSI = E ref + E PT2 . (3)
Finally, all the selected determinants are added to the reference space S for the next iteration.

For the generation of different lengths of the CIPSI component of the QMC wave function, one could either follow (a) an "expansion" scheme where we repeat the above process as many times as necessary, thereby systematically increasing the size of the wave function, or (b) a "truncation" scheme after generating a very large CIPSI wave function, typically to a point where the CIPSI energy (Eq. 3) is reasonably converged. The large wave function created for the second scheme can contain millions of determinants and one needs to reorder them in decreasing order of the absolute value of their CI coefficients and then truncate the expansion, keeping their coefficients the same as in the large CIPSI wave function. This strategy is said to be a better starting point for VMC optimization as the determinant coefficients are from a near FCI calculation 25 . We employ the expansion scheme while keeping the wave function an eigenstate of S 2 while no such constraint is used when following the truncation scheme. Hence, the comparison of the two schemes is not exactly one to one. We use complete active space SCF (CASSCF) orbitals for the majority of these calculations, however, (c) the use of natural orbitals obtained from a large CIPSI calculation is another aspect of our investigation.

After generating the CIPSI expansion, we introduce a positive Jastrow correlation factor J and construct the Jastrow-CIPSI wave function as

ψ = J ψ CIPSI = J N det i=0 c i D i , (4)
where J explicitly describes electron-electron and electronnucleus (two-body) and electron-electron-nucleus (threebody) correlations [START_REF] Filippi | [END_REF] while imposing the electron-electron cusp conditions. N det represents the number of determinants in the CIPSI wave function spanning the space S.

III. COMPUTATIONAL DETAILS

The QMC calculations are carried out with the program package CHAMP [START_REF] Umrigar | Champ: Cornell-holland ab-initio materials package[END_REF] . We employ scalar-relativistic energyconsistent Hartree-Fock pseudopotentials and the cc-pVTZ Gaussian basis set specifically constructed for our pseudopotentials [START_REF] Burkatzki | [END_REF][START_REF]For the hydrogen atom, we use a more accurate BFD pseudopotential and basis set[END_REF] . In particular, we perform all calculations with the cc-pVTZ basis set and test the convergence of the results with the cc-pVQZ basis set. To generate the Slater component of the QMC wave functions, CIPSI calculations are performed in Quantum Package 37 using canonical orbitals obtained from a CAS(10,10) MCSCF calculation for the ground state of butadiene carried out with the program GAMESS(US) [START_REF] Schmidt | [END_REF][START_REF] Gordon | Theory and applications of computational chemistry[END_REF] .

All parameters (Jastrow, orbital, and CI coefficients) are variationally optimized in VMC using the stochastic reconfiguration method [START_REF] Neuscamman | [END_REF] in a conjugate gradient implementation 41 . Most calculations are performed with a two-body Jastrow factor and the impact of the electron-electron-nucleus terms on the energy is tested for a few cases. Exploiting the lownumerical-scaling computation of energy and wave function derivatives 2 , the ground-state geometry of butadiene is simultaneously optimized with the wave function following the path of steepest descent and an appropriate rescaling of the interatomic forces. To avoid spikes in forces, an improved estimator of the inter-atomic forces is used in all calculations, which is obtained by sampling the square of a modified wave function close to nodes 42 . In the DMC calculations, we treat the pseudopotentials beyond the locality approximation 43 and use an imaginary time step of 0.015 a.u. As shown in the Supplementary Information (SI), this time step yields DMC energies converged to better than 0.1 mHartree for a simple Jastrow-CIPSI wave function with only two determinants and is therefore appropriate for all wave functions of higher quality considered in this work.

We employ Gaussian09 [START_REF] Frisch | Gaussian[END_REF] to perform CCSD(T) geometry optimization in combination with our pseudopotentials and in all-electron calculations in the frozen core (FC) approximation with the cc-pVXZ (X = D, T, Q, and 5) and aug-cc-pVXZ (X = D, T, and Q) basis sets. For CCSD(T) geometry optimization without the FC approximation, we use the the PSI4 code [START_REF] Parrish | [END_REF] with the cc-pCVXZ, cc-pwCVXZ, and corresponding augmented (X = D, T, and Q) basis sets. The results of these optimizations and their extrapolations to the CBS limit are detailed in the SI.

IV. RESULTS AND DISCUSSION

We investigate the merits of the choice of a CIPSI expansion to describe the Slater component of a QMC wave function when optimizing the ground-state geometry of butadiene over the conventional CAS description. We start all structural optimizations with the Jastrow-CIPSI wave functions from the same initial MP2/cc-pVQZ optimized geometry and, as described above, variationally optimize the Jastrow parameters, CI coefficients, and orbitals simultaneously with the geometry in VMC. Post convergence of the VMC energy and stabilization of the bond lengths, we perform 40 additional iterations and average these geometries. A final VMC and DMC energy calculation is done on this average geometry with the wave function obtained in the last iteration. As detailed in the SI, reoptimizing the wave function on the average geometry leads to equivalent energies within the statistical error.

We primarily focus our discussion on the use of the "truncation" scheme for the CIPSI-determinant selection since, unlike the expansion scheme, no added selection criterion has been used. An initial CIPSI wave function is constructed including as many as 1.17 million determinants and then truncated to generate a set of determinantal expansions of increasing size. The results are summarized in Table I and illustrated in Figs. 1 and2, where the final VMC and DMC energies, and corresponding structural parameters are plotted against the size of the CIPSI expansions. We also compare our results with previous QMC calculations 2 of the structural optimization of butadiene, which employed various CAS expansions: a CAS(4,4), CAS(4,16), and CAS(4,20) correlating 4 π electrons in the orbitals constructed from the 2p z , 3p z , 3d xz , 3d yz , and 4p z atomic orbitals; a CAS(10,10) consisting of 6 σ and 4 π electrons in 10 bonding and antibonding orbitals; a truncated RAS (10,22) that includes single and double excitations in additional 12 π and δ orbitals over the CAS (10,10) space. Since the cc-pVTZ and cc-pVQZ basis sets yield VMC and DMC energies compatible to better than 2 and 0.2 mHartee, respectively, and structural parameters differing by less 0.4 m Å (see SI), we can directly compare our cc-pVTZ calculations with the results obtained with the optimization of Jastrow-CAS wave functions in the cc-pVQZ basis.

As shown in Fig. 1, a CIPSI wave function with roughly 100 determinants yields a VMC energy which is only about 1 and 4 mHartree higher than the energies of the CAS(10,10) and RAS (10,22) expansions consisting of 15,912 and 45,644 Slater determinants, respectively. Unlike the Jastrow-CAS case, where the energies are scattered around a relatively flat value, CIPSI expansions of increasing size yield a monotonic decrease in the VMC energy, with our largest considered expansion of 32,768 determinants amounting to an energy which is about 40 mHartree lower than the best RAS (10,22) value. In fact, the VMC energy obtained with our largest CIPSI expansion is within 5 mHartree of the converged DMC energy obtained with the Jastrow-CAS wave functions. Therefore, a smart selection of determinants from the approximate FCI space helps us attain much lower VMC energies in comparison to CAS expansions over conventionally used active-space definitions, which instead lead to the inclusion of many determinants with little contribution to the energy.

The behavior of DMC parallels the VMC results with the energy of the Jastrow-CAS wave functions being lowered by about 8 mHartree when the size of the wave function is increased from 1 to 20 determinants, and stagnating afterwards as shown in Figure 1. The Jastrow-CIPSI wave function yields a DMC energy which is comparable with the Jastrow-CAS case when the CIPSI expansion has only about 100 determinants, while the largest CIPSI expansion gives a DMC energy 12 mHartree lower than the truncated RAS (10,22) case. We also note that the estimate of the FCI limit in the current basis set on the initial geometry is about -26.275 Hartree (see SI) and, therefore, as much as 37 and 54 mHartree higher than our best VMC and DMC energies. Our best DMC energies are also superior to the CCSD(T) optimized values obtained with a quintuple-ζ basis set (see SI). Importantly, the use of CIPSI expansions (in combination with a Jastrow factor and the optimization of all wave function parameters) is not only beneficial in terms of the quality of the final total energies: the smooth and monotonic convergence of the VMC and DMC energies with the number of determinants demonstrates the effectiveness of such an approach in identifying energetically relevant determinants in a systematic manner. These important excitations are not easily accessible through manual selection as demonstrated by the FIG. 1. Convergence of the VMC and DMC energies on the optimal geometry with the number of determinants in the CIPSI expansion (red). Previous results 2 obtained with Jastrow-CAS wave functions are also presented (blue).

energy plateau one reaches in constructing very large expansions based on an a priori choice of an apparently reasonable set of active orbitals. Besides being completely automated, this feature of the CIPSI scheme is also crucial for obtaining a smooth convergence of the structural properties (to better than 1 m Å) with the number of determinants as discussed next.

1.446 In Figure 2, we illustrate the variation of the carbon-carbon single and double bonds and of the bond length alternation (BLA) obtained with the Jastrow-CIPSI expansions. The convergence of the bond lengths and BLA is smooth with the increase in the size of the wave function. While the double bond is already converged to better than 0.5 m Å with little over 2000 determinants, the single bond and, consequently, the BLA show a somewhat slower convergence and reach the same level of accuracy with about 5114 determinants. We stress however that we are here looking at extremely small differences while establishing the degree of convergence. For all practical purposes, the BLA is already converged to better than 1 m Å within the limits of statistical error with just over 1000 determinants, again proving the ability of CIPSI expansions to obtain optimal geometry descriptions with a very small set of determinants. Besides overcoming the nonuniform convergence of the structural parameters when expanding the wave function on different active spaces, these calculations also confirm the need for a rather subtle multireference description for such conjugated systems 46,47 . The converged single bond length is in excellent agreement with the CCSD(T)/CBS value of 1.455 Å obtained with the same pseudopotentials used in the QMC calculations (see SI). Our double bond is instead less than 0.003 Å smaller than the corresponding CCSD(T)/CBS value, consequently resulting in a difference of about 0.002 Å on the BLA.

To investigate the impact of the inclusion of three-body terms in the Jastrow factor, we add them to the CIPSI wave functions with 2048 and 5114 determinants and reoptimize all wave function parameters on the fixed average geometries we have determined with the two-body Jastrow factor. These optimizations result in the VMC and DMC energies presented in Table II. While there is an expected gain in the VMC energies, the DMC energies are equivalent within the statistical error of 0.2 mHartree to the energies obtained with a two-body Jastrow factor. Consequently, given the quality of our determinantal component, a two-body Jastrow recovers most of the missing dynamical correlation contribution and is sufficient for our purposes. All results presented so far have been obtained by selecting the CIPSI determinants out of a much larger CIPSI wave function according to the "truncation" scheme. The VMC energies obtained with the "expansion" scheme where one constructs CIPSI expansions of increasing size are compared with the "truncation" scheme in Figure 3. Both schemes lead to very comparable convergence in the energy as a function of the number of determinants in the Jastrow-CIPSI wave function and to compatible converged structural parameters. However, as shown in the SI, the variation of the bond lengths and BLA is not as smooth as in the truncation case. This differ- CAS CIPSI "expansion" CIPSI "truncation" FIG. 3. Optimal ground-state VMC energies obtained with Jastrow-CAS and Jastrow-CIPSI wave functions constructed with the "expansion" and "truncation" schemes. ence probably arises from the added requirement of having an eigenstate of S 2 in the expansion scheme. In fact, if we follow the expansion scheme without this requirement and construct a wave function of 5114 determinants, we obtain an optimal VMC single and double bonds of 1.45564(26) and 1.33496(11) Å respectively, and, a BLA of 0.12068(26) Å. The VMC energy converges to -26.29960(24) a.u. These results are in good agreement with the corresponding values obtained with the truncation scheme.

Finally, we explore a potentially better orbital description in terms of natural orbitals obtained from a CIPSI calculation instead of the canonical orbitals resulting from a CAS(10,10) self-consistent calculation as done above. To this aim, we determine the natural orbitals for a very large CIPSI wave function of 2 million determinants and construct three expansions of roughly 2000, 5000, and 15,000 determinants with the truncation scheme. Surprisingly, we consistently obtain higher VMC energies compared to the expansions generated with canonical orbitals. The reason behind this observation is unclear. The bond lengths and BLA obtained with 5000 determinants are in excellent agreement with the corresponding values for the truncated canonical set of comparable size but the other test cases do not provide as compatible values. The results of these calculations are given in the SI.

V. CONCLUSION

We demonstrated the excellent performance of compact perturbatively-selected CIPSI determinantal expansions in obtaining significantly lower VMC and DMC energies as compared to conventional active space definitions for the challenging case of butadiene. With the use of these wave functions, we were able to obtain converged ground-state structural parameters with the use of only a few thousand determinants. We tested two different schemes for the selection of the CIPSI expansions either by constructing a large CIPSI wave function and then truncating it ("truncation" scheme), or by considering successive sets of determinants in the CIPSI construction ("expansion" scheme). We found that the two representations are rather equivalent in terms of energy but that the truncation scheme possibly leads to a somewhat smoother convergence of the structural parameters with the size of the CIPSI expansion. CCSD(T) calculations with the same pseudopotentials yield a CBS estimate of the carbon-carbon single bond in very good agreement with our converged value but a double bond and corresponding BLA smaller by about 0.002 Å. We do not expect any significant change in the bond lengths upon inclusion of additional determinants, in view of their weak variations observed over a wide range of the number of determinants already considered.

Our study therefore shows that the automated selection of determinants from a CIPSI wave function is an extremely suitable and less cumbersome alternative for the fast optimization of ground-state geometries in QMC than a choice based on correlating electrons in active spaces constructed from energetically low-lying orbitals. The latter results in large expansions with many determinants which contribute little to the energy and to the convergence of the structural parameters. The use of CIPSI-based wave functions in combination with the low-scaling algorithms for simultaneous wave function and geometry optimization opens the way to the accurate and efficient QMC optimization of large molecular systems.

FIG. 2 .

 2 FIG. 2. Convergence of the single bond (C-C), double bond (C=C), and bond length alternation (BLA) ( Å) with the number of determinants in the CIPSI wave function (red). Previous results 2 obtained with a Jastrow-CAS wave function are presented in blue.

TABLE I .

 I Optimal ground-state structural parameters ( Å) of butadiene and corresponding VMC and DMC energies (a.u.) with increasing number of CIPSI determinants obtained in the truncation scheme. The total number of optimized parameters in the wave function is listed. The statistical error is given in brackets.

	No. det No. param	C-C	C=C	BLA	VMC	DMC
	1	749	1.45595(28) 1.32415(11) 0.13180(30) -26.24310(32) -26.30426(28)
	2	782	1.44596(20) 1.33025(14) 0.11571(29) -26.24912(32) -26.30681(14)
	8	822	1.45244(28) 1.33245(15) 0.11999(50) -26.25644(31) -26.31044(26)
	128	1594	1.45778(12) 1.33564(20) 0.12214(20) -26.26562(30) -26.31223(24)
	1024	5514	1.45632(22) 1.33493(08) 0.12139(25) -26.28829(26) -26.31908(20)
	2048	7726	1.45626(15) 1.33456(09) 0.12170(17) -26.29386(25) -26.32147(16)
	5114	12147 1.45549(08) 1.33434(11) 0.12115(07) -26.29980(24) -26.32424(09)
	15469	24818 1.45491(06) 1.33406(08) 0.12085(10) -26.30880(22) -26.32873(22)
	32768	44265 1.45487(25) 1.33414(21) 0.12072(37) -26.31194(12) -26.32928(20)

TABLE II .

 II Effect of the inclusion of three-body Jastrow terms on the total VMC and DMC energies (a.u.). The structures optimized with the two-body Jastrow factor are used. ∆E denotes the gain in energy with respect to the values obtained with a two-body Jastrow factor.

	No. det	EVMC	EDMC	∆EVMC	∆EDMC
	2048 -26.29908(15) -26.32162(20) -0.00522(29) -0.00015(26)
	5114 -26.30333(23) -26.32412(16) -0.00353(33) 0.00012(18)

ACKNOWLEDGEMENTS

This work is part of the Industrial Partnership Programme (IPP) "Computational sciences for energy research" of the Netherlands Organisation for Scientic Research (NWO-I, formerly FOM). This research programme is co-financed by Shell Global Solutions International B.V. This work was carried out on the Dutch national supercomputer Cartesius with the support of SURF Cooperative, and using HPC resources from CALMIP (Toulouse) under allocation 2017-0510. The authors declare no competing financial interest.

SUPPLEMENTARY INFORMATION

See Supplementary Information for CIPSI energies and FCI extrapolation; VMC and DMC energies of bare CIPSI wave functions; impact of reoptimizing the wave function on the average geometry; basis-set convergence of geometry and corresponding VMC and DMC energies; bond-length and BLA convergence obtained with the "expansion" scheme; energy and geometry convergence with the use of natural orbitals; DMC time-step extrapolation and CCSD(T) geometry optimizations with pseudopotential and all-electron basis sets.


Delphine Fradin

Pascal Boileau

Jacques Lepercq

Pierre Bougneres
email: pierre.bougneres@wanadoo.fr

MD P Bougnères

'Non-Mendelian' genetics of fetal growth

Keywords:

come

MENDELIAN GENETICS OF FETAL GROWTH

Mendelian genetics, ie the genetics of monogenic diseases, have become an important field of pediatric endocrinology. The restriction of fetal growth offers examples of Mendelian diseases (small size for gestational age with monogenic causes). To induce pathological manifestations, mutations have to alter a gene product necessary to a function or a pathway which is itself necessary to allow normal fetal growth. The alteration of the gene product is generally qualitative (a truncated or a functionally deficient protein), more rarely quantitative (a markedly reduced expression of the protein of interest). Nonsense or missense mutations located within exons or the deletion of useful exons can result in abnormal proteins that have lost most, if not all of their function. Promoter mutations or epimutations modifying DNA methylation can result in decreased expression of the protein.

Understandably, evolution has tried to eliminate disabling mutations as 'errors of nature'. For this reason, most monogenic diseases and their causative mutations are rare. Monogenic disorders can sometimes be mimicked by the experimental manipulation of the gene of interest, by altering its RNA or protein products in a knock-out or knockin mouse, or in a cell line.

GENETIC VARIATION OF FETAL GROWTH THAT IS DEPENDENT ON IMPRINTING

Genomic imprinting is an epigenetic process that marks the parental origin of certain genes. Imprinting results in the monoallelic expression of a gene, at least for a given time (intrauterine life for example) and in a given tissue. The monoallelic expression depends on the parental origin of the allele. Genes that are not expressed from the maternal allele are said to be maternally imprinted (ie, maternally imprinted genes are those whose paternal allele is expressed alone). For most genes following the rules of Mendelian genetics character transmission, the parental origin of genetic information has no influence on gene expression. In contrast, genomic imprinting follows the rules of 'parent-specific gene expression'. Most imprinted genes are found clustered with other imprinted genes. This clustering of imprintable genes is likely to have functional consequences that are not yet understood. As many of the known imprinted genes are involved in the control of fetal growth, genomic imprinting might be a key component of the mechanisms underlying maternal constraint, which limits fetal size in order to allow vaginal delivery. The first observation of an imprinted gene was obtained by the knockout of the Igf2 gene by homologous recombination in mice [START_REF] Dechiara | Parental imprinting of the mouse insulin-like growth factor II gene[END_REF]. Fetal growth was reduced if the inactivated allele was transmit-ted paternally, but normal when the maternal allele was mutated. Conversely, maternal inheritance of a non-functional Igf2r gene results in fetal overgrowth, whereas paternal transmission of the disrupted allele leads to normal fetal size [START_REF] Lau | Loss of the imprinted IGF2/cation-independent mannose 6phosphate receptor results in fetal overgrowth and perinatal lethality[END_REF]. Recently, Constancia et al. [START_REF] Constancia | Placentalspecific IGF-II is a major modulator of placental and fetal growth[END_REF] have mutated the placenta-specific promoter of Igf2 gene and their results were comparable to the complete knockout of Igf2 gene [START_REF] Constancia | Placentalspecific IGF-II is a major modulator of placental and fetal growth[END_REF]. Newborn mice with a paternal copy of the mutated promoter showed comparable growth retardation to mice with a complete disruption of the Igf2 gene. This indicates that a paternally expressed (maternally imprinted) gene in the placenta acts as a regulator of mouse fetal growth. This example shows that imprinted genes imply a differential role for maternally vs paternally derived genetic information, and may have different consequences on placental function. Knockouts of the paternally expressed Igf2, Peg1 and insulin genes result in fetal growth restriction, whereas knockout of the maternally expressed Igf2r gene results in fetal overgrowth. These results support the prediction that paternally expressed (maternally silenced) genes enhance fetal growth, whereas maternally expressed (paternally silenced) genes act as growth inhibitors [START_REF] Reik | Genomic imprinting: parental influence on the genome[END_REF]. It is fascinating that the maternal environment (nutrition, climate, light, workload, etc) could be perceived by the pregnant mother's body and generate signals that induce chemical modifications of parts of the fetal genome which are involved in regulation of fetal growth [START_REF] Gluckman | Maternal constraint of fetal growth and its consequences[END_REF]. In humans, fetal growth restriction can result from alterations in expression of imprinted genes. A major cluster of imprinted genes occurs at chromosome 11p15. This cluster contains IGF-2 and H19 genes, which are reciprocally imprinted, and several genes which are under the control of a differentially methylated CpG island (KvDMR1). Molecular defects affecting these genes are involved in the Beckwith-Wiedemann syndrome (BWS). BWS is a syndrome of pre-natal overgrowth with macroglossia, hypoglycemia and visceromegaly. Paternal uniparental disomy for 11p15 is found in 20% of BWS patients [START_REF] Henry | Uniparental paternal disomy in a genetic cancer-predisposing syndrome[END_REF]. In addition, the bi-allelic expression of IGF2 has been associated with somatic overgrowth in children having no other features of BWS [START_REF] Morison | Somatic overgrowth associated with overexpression of insulin-like growth factor II[END_REF]. More recently, an epimutation has been described in some patients with Silver-Russell syndrome (SRS) [START_REF] Gicquel | Epimutation of the telomeric imprinting center region on chromosome 11p15 in Silver-Russell syndrome[END_REF]. SRS is a syndrome of prenatal growth retardation with body asymmetry and a small triangular facies. A demethylation (epimutation) was identified in the imprinting centre region ICR1 of this 11p15 region. This demethylation is responsible for the biallelic expression of H19 and the downregulation of IGF-2 expression that probably results from a relaxation of imprinting. Ten percent of patients with BWS have a hypermethylation of the H19 promoter [START_REF] Gaston | Analysis of the methylation status of the KCNQ1OT and H19 genes in leukocyte DNA for the diagnosis and prognosis of Beckwith-Wiedemann syndrome[END_REF]. This molecular mechanism is the opposite of the epimutation described in SRS patients. These observations suggest that paternally expressed and maternally silenced IGF-2 is an important regulator of human fetal growth. The effect of genomic imprinting is the expression of only one copy of a gene during fetal growth. This form of gene control allows the other gene copy to remain available eitherfor critical periods of rapid post-natal growth or for functioning during tissue maturation. The predominant theory for genomic imprinting is known as the 'parental conflict theory', which focuses on the nutritional needs of the fetus that are met by the mother via the placenta [START_REF] Wilkins | What good is genomic imprinting: the function of parent-specific gene expression[END_REF]. This theory leads to the prediction that imprinted growth enhancers will be silent in maternal derived alleles whereas imprinted growth inhibitors will be silent in paternal derived alleles. This theory has been successful at explaining the imprinting of paternally expressed enhancers (IGF-2, KCNQ1OT1) and maternally expressed inhibitors (Igf2r, CDKN1C) of pre-natal growth. Because disrupted imprinting genes can lead to phenotypic changes in pre-natal growth, it is likely that genetic determinants of fetal size may be found in imprinted genes. It is suspected that genomic variants located within imprinted genes will be identified as one of the mechanisms underlying fetal growth variation. However, low birth weight children have higher mortality not only soon after birth but also later on in their adult life. Therefore, it is important to keep in mind that imprinting, although not the only mechanism involved in epigenetic processes, could also interact with the programming phenomena that link fetal life to post-natal environmental conditions. Changes in expression of imprinted genes could also provide a genetic link between intrauterine growth and the programming of fetal systems involved in the long-term risk of metabolic disease in adulthood, notably in the field of insulin resistance.

NON-MENDELIAN GENETICS

Non-Mendelian genetics belong to a different world. They deal mostly with the question of how genomic variations contribute to individual differences, ie the phenotypic variation. In case of quantitative traits, people cannot be categorized as 'cases' or 'controls', they are human individuals with phenotypic variability. The extensive polymorphism of the human genome, which results in varying protein isoforms and varying protein quantities in various tissues explains much of the variation of each human trait. Humans do not have, for instance, the same birth weight, adult height, glucose level, blood pressure, muscle

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FETAL GROWTH: A MULTIFACTORIAL MULTIGENIC TRAIT

Fetal growth is without doubt a complex trait (Fig. 1), with a partial genetic determination itself regulated by many genes. The existence of many genes important to fetal growth is demonstrated by the vast number of chromosomal abnormalities that are capable of impairing fetal growth severely. A fraction of the genes regulating fetal growth has functional SNPs (coding or regulatory). Functionally important SNPs or their variants can also be located outside of genes in the intergenic non coding DNA portions of the genome. Whether they are coding, interfering with splicing, or regulating gene expression, functional genomic SNPs should not be considered as deleterious mutations. There is no disease here, no 'errors of nature', only the natural variation of fetal growth. Nature has used genomic SNPs to shape and transmit a degree of diversity of fetal growth across individuals. To be expected to play a significant role in fetal growth at the population level, a SNP needs (a) to affect the trait in a quantitatively sufficient manner (affect size), and (b) to be present in a rather large fraction of people, ie, its allelic frequency should exceed 0.10-0.20. To be frequent in the modern human population, a SNP has to be ancient, and retained and expanded throughout human history (possibly because it has been useful in a given environment). It is likely that fetal growth has been the subject of genetic selection during evolution, because of its importance for survival. Birth is clearly a critical moment of life, maybe the most critical. The perinatal period is characterized by the highest mortality risk that a human being has to face during his entire life. Size at birth seems to play a major role, as smaller or larger than normal babies have increased mortality. Evolution has stabilized human birth size within relatively narrow limits in order to reduce perinatal mortality. There is however a degree of interindividual variability. Genes important for the physiology of fetal growth can certainly be unravelled in animal models, at least for the part of physiology that is not species specific. In contrast, animal models are unlikely to tell us which human SNPs are important to shape the variations of human fetal size, because these SNPs are human. Genomic variants that determine the variations of fetal growth can only be searched for in humans. Reimplanting them through genetic manipulations in mice to demonstrate their precise physiology will be difficult, because the penetrance of their expression will have to go through the mice background (both genetic and environmental).

FETAL GROWTH, A PHENOTYPE MADE OF SEVERAL PARTS

Genetics studies the relationships of genes (or SNPs located outside genes) with a given phenotype. Fetal growth is not a simple homogeneous phenotype.

The fetus is made of several components that are likely to have their own genetics. The quantitative trait, referred to as fetal size, measured at birth in g and cm, has to be dissected in several 'sub-traits' or components: body mass including all organs, fat and Fig. 1 -From genes to complex traits.

Trait

Complex trait

Components of complex trait

Physiology Environment

Functional pathways

Gene products varying in structure or abundance +

Gene products without individual variation

Gene expression Environment

Genes with or without functional variants

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HOW MANY GENES ARE EXPECTED TO BE IMPORTANT FOR VARIATIONS IN FETAL SIZE?

Fetal growth is a multifactorial phenomenon that is dependent on the maternal environment and on a panoply of fetal genes. But only a fraction of these genes are subjected to variation, ie, those which carry functional SNPs interfering with growth related physiology. The heritability of birth weight and height is around 0.50, indicating a significant role of genetic factors [START_REF] Morton | The inheritance of human birth weight[END_REF]. It is yet impossible to know how many genes are involved in the variation of these traits. For a long time, geneticists have thought that complex traits follow the 'infinitesimal model' which proposes that a large number of genes (hundreds), each playing a minuscule role (<1/100 th of the overall variance), are involved in trait variation. It seems more plausible today that even very complex traits have an oligogenic control involving a few 'major' variants [START_REF] Farrall | Quantitative genetic variation: a post-modern view[END_REF], each of them accounting for 5-10% of the variance of the trait. There are no indications yet of the number of variant alleles determining the natural variations of fetal growth in any mammal.

A GENETIC APPROACH OF FETAL HEIGHT

In a recent study, our group obtained birth height measurements from hundreds of children born at term to Caucasian European women after un-eventful gestations and deliveries. Birth height was considered as a quantitative parameter following a continuous distribution (Fig. 2). Based on classical genetic data and evaluation of heritability around 0.50, it was hypothesized that 'non-Mendelian' genetics were playing a role in the variation of human birth height. There were two options to identify the genes and variants that play a role in the individual variation of birth height. The first one is to list genes known for their role in the physiology of human fetal growth (or mammalian fetal growth) and to find the known SNPs within or near these genes using data bases or sequencing of these regions. These SNPs could a priori be suspected to be functional genomic variants (Gf) important to growth of fetal skeleton or to be closely linked as markers (Gm) to the functional SNPs (Gf). Finally the statistical association of fetal size with these SNPs, or blocks of SNPs (haplotypes) can be tested. A second option is to have no a priori hypothesis regarding genes responsible for the skeletal growth of human fetuses, and to engage in a 'blind' genomewide approach. This requires the use of genomic markers (Gm), hundreds of microsatellites or tens or hundreds of thousands of anonymous SNPs scattered on the whole human genome. Hundreds of pairs of siblings are needed for this purpose. The phenotypic variation of fetal height can be modelled as the intra-pair difference between the birth heights of two siblings, then tested against the similarity of genomic variation at given polymorphic sites on the genome (microsatellites or SNPs) of the two siblings.

Our study group decided to follow the first scenario, which requires the study of 400 microsatellites spanning the whole genome. At each microsatellite site, the studied siblings share 0, 1 or 2 alleles identical by descent from their parents. A variance compo- nent method can be used to study the relationship between the intra-pair sharing of microsatellite alleles and the observed intra-pair difference of birth height. Using this approach, we found that two chromosomal regions displayed a high probability of being associated with birth height variation. These regions can be considered as 'fetal height quantitative trait locus' (QTL). Because these regions are very large (3-10 Mb) and contain hundreds of potential candidate genes, a fine-mapping approach has to be used. This method uses a dense collection of genomic markers in the regions which we previously identified as QTL. Fine mapping will keep the candidate genes to a manageable number. These genes will then be tested. The ultimate aim is the identification of genomic variations [QTNs: quantitative trait nucleotides(QTN)] which are within or near these genes, and which are associated with variations of human height at birth.

CONCLUSIONS

Scientific evidence generated from Mendelian genetic studies can explain to some extent the genetic origins of certain growth related disorders. It is documented that fetal growth restrictions can result from alterations in expression of imprinted genes and that the major cluster of such genes occur at chromosome 11p15. Non-Mendelian genetics, however, seems to be more capable of explaining phenotypic variability. Fetal growth is a complex multifactorial, multigenic trait. This is also suggested by the large number of chromosomal abnormalities which play a role in impaired fetal growth. Evolution has stabilized human birth size within relatively narrow limits in order to reduce perinatal mortality, but with a degree of individual variability. The quantitative trait of fetal size is comprised of several 'sub-traits' or components: body mass including all organs, fat and muscle, brain mass, head circumference, spine and limb skeletal growth. It is likely that different genetic factors and genomic variants are responsible for the variations of these 'sub-traits'. A study has been initiated by our group to test the hypothesis that non-Mendelian genetics plays a role in the variation of human birth weight. The ultimate aim is the identification of genomic variations that are within or near certain selected candidate genes and are associated with variations of human height at birth.

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  Signature ...................................................................

	Non-Mendelian genetics of fetal growth
	or fat mass. They owe these differences in part to the
	physical and cultural environment in which they live,
	but also to their specific genomic constitution, which
	is different in every individual. The human genome
	contains approximately 10 million single nucleotide
	polymorphisms (SNPs), each of them with two alle-
	les generating three genotypes (or diplotypes). The
	10 millions single nucleotide polymorphisms of the
	human genome could in theory generate 3 10.000.000
	combinations of diplotypes. This number is however
	a gross overestimation of human genomic diversity
	because many SNPs are linked to each other on the
	same DNA strand, with their alleles being inherited
	as unrecombined blocks, generating what is called
	linkage disequilibrium between these alleles.
	3

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		Fetal size
		Body	Body
		height	weight				Fig. 2 -Fetal size dissected in its compo-
	Limb length	Head circumference Spine length	Brain mass	Fat mass	Muscle mass	Tissues, organs	nents. Genes that have no variant allele associated with variation of fetal size, but are nevertheless important to the determination of fetal size are in open circles. Genes that have functional vari-ants influencing the variation of fetal size
							are shown as closed circles, with those
							having a marked influence on fetal size
						Environment	displayed as larger circles. Gaussian curves are shown to indicate that quanti-
						Genes	tative traits each have their own variation distribution.

Signature ...................................................................


Delphine Fradin
email: fradin@paris5.inserm.fr

P Bougneres

Three Common Intronic Variants in the Maternal and Fetal Thiamine Pyrophosphokinase Gene (TPK1) are Associated with Birth Weight

Keywords: TPK1, birth weight, association

Extreme variations in birth weight increase immediate postnatal mortality and morbidity, and are also associated with the predisposition to metabolic diseases in late adulthood. Birth weight in humans is influenced by yet unknown genetic factors. Since the 7q34-q35 region showed linkage with birth weight in a recent human genome scan (p = 8.10 -5 ), this study investigated the TPK1 (thiamine pyrophosphokinase) gene locus, located in 7q34-36. Having found no coding variants in the TPK1 gene, we genotyped 43 non coding SNPs spanning a region of 420kb, and used the QTDT method to test their association with birth weight in 964 individuals from 220 families of European ancestry. Family-based tests detected association of 8 SNPs with birth weight (p<0.008), but after correction for multiple tests only rs228581 C/T (p = 0.03), rs228582 A/G (p = 0.04) and rs228584 C/T (p = 0.03) were still associated with birth weight, as well as their T-A-T haplotype (p = 0.03). In addition, we found an association between maternal rs228584 genotype and offspring birth weight (p = 0.027).

These observations suggest that genomic variations in the fetal and maternal TPK1 gene could contribute to the variability of birth weight in normal humans.

Introduction

Birth weight is a complex multifactorial trait. It is an important risk factor not only for infant mortality and morbidity [START_REF] Guyer | Annual summary of vital statistics-(1998)[END_REF], but also for adult metabolic disorders [START_REF] Barker | The fetal and infant origins of adult disease[END_REF]) such as obesity, type 2 diabetes (T2D) or cardiovascular diseases, independent of gestational age and of potential confounding factors such as social class or smoking, an observation that has been replicated in several populations from different countries. Birth weight risk associations are not confined to differences between the smallest and large infants, but relate to a continuum of variable risk throughout the whole range of birth weights. Faced with low birth weight or macrosomia, paediatricians often declare them "idiopathic" since environmental and genetic factors remain elusive in the vast majority of cases. The current study is a preliminary step into the genetics of human birth weight variability.

Family and twin studies provide evidence that genetic factors influence birth weight [START_REF] Johnston | Genetic factors contributing to birth weight[END_REF][START_REF] Magnus | Causes of variation in birth weight: a study of offspring of twins[END_REF][START_REF] Magnus | Parental determinants of birth weight[END_REF][START_REF] Morton | The inheritance of human birth weight[END_REF]. A recent genome scan for birth weight in 220 Caucasian families [START_REF] Fradin | Identification of distinct QTLs affecting length or weight variability at birth in humans[END_REF]) indicated the highest degree of linkage in 7q34-q35 with a LOD score of 3.1 (p = 0.00008). According to the stringent linkage criteria proposed by [START_REF] Lander | Genetic dissection of complex traits: guidelines for interpreting and reporting linkage results[END_REF], this result strongly suggests that the 7q34-35 region contains allelic variations influencing normal birth weight variability. Among the more than forty genes located within 7q34-q35, we elected to study TPK1 (thiamine pyrophosphokinase). TPK1 is a cellular enzyme, abundantly expressed in maternal, placental and fetal tissues [START_REF] Bellyei | Cloning, sequencing, structural and molecular biological characterization of placental protein 20 (PP20)/human thiamin pyrophosphokinase (hTPK)[END_REF], which catalyzes the conversion of thiamine, a form of vitamin B1, to thiamine pyrophosphate (TPP).

TPP is an active cofactor for enzymes involved in glycolysis and energy production, including transketolase, pyruvate dehydrogenase, and alpha-ketoglutarate dehydrogenase. Vitamin B1 is an essential vitamin for vertebrates. Our reasons for testing TPK1 as a candidate gene for birth weight control was: i) TPK1 harbours a number of common variants in Caucasians; ii) TPK1 is implicated in thiamine metabolism, and maternal thiamine deficiency causes intra-uterine growth retardation in rats (Butterworth, 1993b;[START_REF] Roecklein | Intrauterine growth retardation induced by thiamine deficiency and pyrithiamine during pregnancy in the rat[END_REF], and children born to mothers suffering from hyperemesis gravidarum, a disease often accompanied by thiamine deficiency, are known to have a decreased birth weight [START_REF] Dornhorst | Management of gestational diabetes mellitus[END_REF][START_REF] Langer | Management of women with one abnormal oral glucose tolerance test value reduces adverse outcome in pregnancy[END_REF].

The TPK1 gene maps to chromosome 7q34-q36, contains at least 9 exons, and spans approximatively 420kb (NM 022445) [START_REF] Zhao | Molecular cloning of human thiamin pyrophosphokinase[END_REF]. According to the db SNP public database, TPK1 has numerous intronic polymorphisms but no coding variants.

The current study analyzed sequence variations of the human TPK1 gene and performed family-based and haplotype-based association tests to determine whether genetic polymorphisms within this gene could contribute to the normal variability of birth weight in humans.

Materials and Methods

Cohort

The study design and recruitment procedures have been published before [START_REF] Fradin | Identification of distinct QTLs affecting length or weight variability at birth in humans[END_REF]. Briefly, inclusion criteria were: 1) a birth weight at term exceeding 2000g (10 patients excluded) and below 5000g (4 patients excluded); 2) gestational age strictly between 39 to 41 weeks, determined by reference to the last menstrual period; 3) healthy gestation (mothers were considered healthy if they had no known diabetes, hypertension, medical condition or addiction associated with impaired fetal growth); 4) access to at least one sib; 5) parental data and DNA available. The study population included 220 families, 154 pedigrees of 2 sibs, 53 pedigrees of 3 sibs, 9 pedigrees of 4 sibs, 3 pedigrees of 5 sibs and 1 pedigree of 6 sibs, leading to a total of 412 sib-pairs. All participants were French Caucasian of European origin. The studied sample included 964 individuals who showed the same distribution of birth weight as the French general population (440 parents, 240 male children and 284 female children). Our cohort can therefore be considered a representative healthy sample of the population living in a developed European country. Term and weight measurements were obtained from the Carnet de Santé completed by maternity paediatricians at birth. French children have to be precisely measured at birth to be included in the obligatory follow-up system of "Carnet de Santé de l'Enfance" (Child Health Bulletin). The study was approved by the Cochin University Institutional Review Board. All of the participants signed an informed-consent document.

Sequencing

We sequenced the complete TPK1 coding sequence (NM 022445) in ten individuals randomly selected from our cohort. Primers used to amplify each TPK1 exon are listed in Table 1. Amplification was carried out in 96 well microtiter plates (Abgen) : each 50µl reaction contained DNA (200ng), MgCl 2 (1.5 mM), 1X PCR Buffer (Invitrogen), dNTPs (0.2 mM each), primers (1 µM each) and Taq polymerase (1.25 units, Invitrogen). PCR products were purified using a QIAGEN kit before sequencing. PCR products were sequenced using a BigDye Terminator v3.1 sequencing standard kit. All sequence determinations were performed in duplicate.

Genotyping

Forty-three SNPs TPK1 were selected from public databases (HapMap project) based on the following criteria : 1) reported to dbSNP by various sources; 2) validation status in Caucasians; 3) high degree of heterozygosity (Table 2, Figure 1). These 43 SNPs were successfully genotyped by Illumina technology in all individuals (parents and offspring, n = 964). The Illumina Golden Gate Assay employs highly multiplexed, allele-specific extension and ligation

1 5 ACGAAGCCTTGGAGACTTGA3 5 CTATGAGGTCCGAGGGGTTG3 2 5 CTCCCGTGCTGTTGGTTC3 5 AAAATGAGACCGGCGAGAG3 3 5 GGACCCGTTTATGTATATGC3 5 CCACTTTTTAGTCAGGTTTCAGG3 4 5 TAGCGTGACTCTGTGCGTCT3 5 AACATGGTTAAGCAGCTCAGG3 5 5 TGCACTTGGCCACAGTTTTA3 5 TTTGTGAAAAAGTGTAAGTTTCCAT3 6 5 TGTCCCATGTGAAATAAAAA3 5 CATCCTTGGAATAACCACCAA3 7 5 GGTTCTGTTGGTGCTTGGTT3 5 TTGTGAAGGAAAGGGCTCAC5 8 5 TTAAGTCGGGTGAAGGTTCC3 5 TGACCCAGCAATTCCTCCTA3 9 5 CCCATTTTAAAGCCATACATCA3 5 CCCCTTCCCCCAATATTTA3 5 GGTTGGAACCTAATGGGAATC3 5 GAGTAGGGCTCCACGAATCA3 5 TGATTCGTGGAGCCCTACTC3 5 TCAGCCCTATGACTAAGTG3 5 GCAGTCATAGGGCTGATAATACA3 5 TCTCACACTGTCATAGCACCAA3 74bp

Haplotypes Reconstruction

Merlin software (Abecasis et al. 2000a) was used to analyze linkage disequilibrium (LD) and reconstruct rs228581-rs228582-rs228584 haplotypes based on data obtained from the studied families. Information about gene flow in a pedigree can be used to reconstruct the more likely haplotypes for families and individuals. The physical distances of the genetic markers required for haplotype reconstruction were derived from the Genethon map.

Statistical Analyses

The Mann-Whitney test and the Kruskal-Wallis analysis of variance by rank were applied to compare, respectively, two and several independent samples when a classical t-test or an ANOVA test could not be used. Data were evaluated using Statview 5.0 (SAS Institute).

Results were corrected for multiple tests by Bonferroni correction. Values are reported as means ± sd.

Quantitative Transmission Disequilibrium Test (QTDT)

Birth weight was analyzed as a continuous trait adjusted for sex. To test for both linkage and association with birth weight variation, we used the QTDT method (Abecasis et al. 2000b) implemented in Merlin [START_REF] Abecasis | Merlin-rapid analysis of dense genetic maps using sparse gene flow trees[END_REF]. The orthogonal model of Abecasis et al.

was used in our analyses, where the total association is partitioned into orthogonal within-and between-family components (ß b and ß w , respectively). The betweenfamily component is specific for each nuclear family and could be confounded by population stratification. However, the within-family component is significant only in the presence of linkage disequilibrium caused by close linkage. Thus, a QTDT test of the significance of ß w based on allelic transmission is robust to

Transcription Element Search System (TESS)

TESS, Transcription Element Search Software (http://agave.humgen.upenn.edu/utess/tess), is a web-based software tool for locating possible transcription factor binding sites in DNA sequence and for browsing the TRANSFAC ® (a database of eukaryotic transcription factors, their genomic binding sites and DNA-binding profiles). It provides functionality beyond that of the TRANSFAC flat files and website. TESS allows the user to search sequence for possible binding sites using either cis-element strings or weight matrix models [START_REF] Fradin | Transcription Element Search Software on the WWW[END_REF].

Alternative Splicing Database (ASD)

The ASD Project aims to understand the mechanism of alternative splicing on a genome-wide scale by creating a database of alternative splice events and the resultant isoform splice patterns of human genes [START_REF] Thanaraj | ASD: the Alternative Splicing Database[END_REF]. Web-based software from the ASD site can be used for intron analysis (donor or acceptor sites, branching points, polypyrimidine tracts etc., http://www.ebi.ac.uk/asd-srv/wb.cgi).

Results

We identified no common coding variants in the TPK1 gene after re-sequencing all known exons in our cohort (Figure 1). We therefore studied a panel of 43 SNPs located within 420kb of the TPK1 gene locus (Table 2). All 43 SNPs were located in intronic regions of the TPK1 gene and all were frequent (m.a.f > 0.10) (Table 2). None of them was located in classical splice donor or acceptor sites. We found that 3/43 SNPs were associated with birth weight after correction for multiple tests: rs228581 (C/T; p = 0.03), rs228582 (A/G; p = 0.04) and rs228584 (C/T; p = 0.03). Allele T of rs228581 was over-transmitted with low birth weight, as well as allele A of rs228582 and allele T of rs228584 (Table 3). We tested a model of dominance for these 3 SNPs and replicated the association between rs228581, rs228582 and rs228584 and birth weight (p = 0.001, p = 0.004 and p = 0.001 respectively). The three associated SNPs are located in intron 3 of the TPK1 gene. D , the linkage disequilibrium parameter, was less than 0.5, supporting the quasi independence of effects due either to rs228581, rs228582 or rs228584.

The reconstruction of haplotypes formed by these three SNPs allowed us to identify only 4 haplotypes (C-A-C, C-A-T, C-G-C and T-A-T, respectively, ordered rs228581, rs228582 and rs228584). Using QTDT we confirmed that the T-A-T haplotype was associated with low birth weight (3362 ± 558g versus 3477 ± 532g for all other pooled haplotypes, p = 0.03).

Since imprinting is an important mechanism of fetal growth regulation, we tested if the parental origin of TPK1 alleles had an effect on the observed association. We found no difference between paternal and maternal alleles of the studied SNPs and haplotypes.

As several maternal genes are known to regulate birth weight, we tested whether variation in the maternal TPK1 genotype was associated with offspring birth weight. We found an association between the maternal rs228584 genotype and offspring birth weight (ANOVA p-value = 0,027), but no association with other SNPs. To differentiate maternal and offspring genotypes in their effect on birth weight, we tested for an association between the untransmitted maternal rs228584 allele and offspring birth weight. We found a significant and relatively large difference in birth weight when the untransmitted maternal allele was C (3283 ± 560g) and when the untransmitted maternal allele was T (3522 ± 500g, p = 0.005). Therefore, the association between

Discussion

Following our genome scan analysis for birth weight [START_REF] Fradin | Identification of distinct QTLs affecting length or weight variability at birth in humans[END_REF] we tested TPK1 as a candidate gene in the 7q34-q36 region. We used a nuclear family-based approach using the QTDT method. The QTDT directly tests for linkage disequilibrium between a trait and a marker locus, and prevents false-positive results in genetic studies due to population stratification. The more commonly used population-based association approaches are open to the risk of spurious associations due to recent admixture or to population samples that are stratified with respect to genetically heterogeneous groups [START_REF] Deng | Population admixture may appear to mask, change or reverse genetic effects of genes underlying complex traits[END_REF]. To avoid these problems the QTDT test uses parental genotypes to construct wellmatched controls. On the other hand, while the linkage approach designed to identify genomic regions harbouring QTLs for continuous traits is often of limited statistical power to detect genes of small effects [START_REF] Risch | The future of genetic studies of complex human diseases[END_REF][START_REF] Sham | Power of linkage versus association analysis of quantitative traits, by use of variance-components models, for sibship data[END_REF], the QTDT is considered much more powerful for testing linkage of specific candidate genes to complex traits [START_REF] Allison | Transmission-disequilibrium tests for quantitative traits[END_REF].

The current sample reflects the continuous distribution of birth weight in normal infants born to healthy Caucasian mothers, with a mean value of 3341g and a standard deviation of 537g, which is comparable to most studies in developed countries [START_REF] Wilcox | Birth weight and perinatal mortality: the effect of maternal smoking[END_REF]. Because mortality augments both sides of the birth weight distribution, evolution has stabilized birth weight symmetrically within a relatively narrow range, around a mean value of 3200-3400g at term [START_REF] Wilcox | Birth weight and perinatal mortality: the effect of maternal smoking[END_REF]. While mean birth weight depends partly on environmental factors, it is expected to depend on a number of genes [START_REF] Magnus | Causes of variation in birth weight: a study of offspring of twins[END_REF][START_REF] Magnus | Parental determinants of birth weight[END_REF]. Each of these can be expected to contribute only a small fraction of the trait variance [START_REF] Farrall | Quantitative genetic variation: a postmodern view[END_REF]. This contribution can be postulated to be 0.5 to 10 percent of the trait variance if the trait is oligogenic [START_REF] Farrall | Quantitative genetic variation: a postmodern view[END_REF], an order of magnitude consistent with the statistical strength of the association observed in the current study.

Using splice sequence analysis we were not able to assign the three intronic SNPs to positions that are known to be critical for the splicing process, for regulation of expression [START_REF] Mizumoto | A genomic element within the third intron of the human oxytocin receptor gene may be involved in transcriptional suppression[END_REF], or for nucleosome formation. The only positive finding was that allele T of rs228581 could bind the N-Oct-3 factor, which belongs to a large family of transcription factors that bind the octameric ATGCAAAT motif. The functional mechanisms through which the three SNPs could affect gene splicing or gene expression thus remains to be established. It is also unknown whether the three SNPs are associated with gain-or loss-of-function of TPK1 enzyme activity, nor are there any physiological data allowing us to document genotypic differences in the thiamine status of the studied mothers or infants.

Because of this association finding, it will be important to examine thiamine metabolism in a prospective genotype-phenotype study. There is an increased requirement for thiamine during pregnancy, which often results in a relative deficiency in the mother [START_REF] Heinze | Determination of thiamine (vitamin B1) in maternal blood during normal pregnancies and pregnancies with intrauterine growth retardation[END_REF]. During normal pregnancies the mean thiamine value in maternal blood cells falls from 230 nmol/l in the 28 th week to 170 nmol/l in the 39 th week [START_REF] Heinze | Determination of thiamine (vitamin B1) in maternal blood during normal pregnancies and pregnancies with intrauterine growth retardation[END_REF]. In cases of severe fetal hypotrophy the corresponding thiamine levels have been found to be 140 nmol/l in the 30 th week and 130 nmol/l in the 39 th week of gestation [START_REF] Heinze | Determination of thiamine (vitamin B1) in maternal blood during normal pregnancies and pregnancies with intrauterine growth retardation[END_REF].

Three key enzymes of glucose metabolism, pyruvate dehydrogenase, α-ketoglutarate dehydrogenase and transkelotase, are thiamine dependent (Butterworth, 1993a;[START_REF] Hsu | Effect of thiamine deficiency on glutathione contents of erythrocytes and tissues in the rat[END_REF][START_REF] Mccandless | Encephalopathy of thiamine deficieny: studies of intracerebral mechanisms[END_REF]). Thiamine deficiency, and consequently TPP deficiency, results in an overall reduction of tissue glucose utilization and impairment of the production of insulin (Rathanaswami et al. 1991;[START_REF] Rathanaswami | Effects of insulin secretagogues on the secretion of insulin during thiamine deficiency[END_REF]1989;1991). The maternal allele T of rs228584 was associated with higher birth weight of offspring in our study. If the TPK1 maternal genotype affects TPK1 enzyme activity, which the current study did not show, it could influence the pool of TPP. Whether it could modify blood glucose and thus influence fetal growth remains entirely speculative.

On the other side, low birth weight infants are at increased risk of metabolic disorders in later life. It is intriguing in this respect that the current 7q34-q36 region has been linked to adult metabolic traits in several studies. Suggestive linkage was found between this 7q region and BMI in 160 pairs [START_REF] Borecki | An exploratory investigation of genetic linkage with body composition and fatness phenotypes: the Quebec Family Study[END_REF]). Significant linkage with BMI to D7S1824 (a microsatellite located on 7q36) was reported in 157 white subjects [START_REF] Platte | A study of linkage and association of body mass index in the Old Order Amish[END_REF], while a LOD score of 2.3 for abdominal fat was found in the same region in 215 subjects [START_REF] Rice | A genomewide linkage scan for abdominal subcutaneous and visceral fat in black and white families: The HERITAGE Family Study[END_REF]. The region 7q35-q36 also harbours a QTL linked to the variation of plasma triglyceride levels (LOD score = 3.7) [START_REF] Sonnenberg | Genetic determinants of obesity-related lipid traits[END_REF]. A LOD score of 1.8 for triglyceride levels was also found in this region of chromosome 7 in another study [START_REF] Shearman | Evidence for a gene influencing the TG/HDL-C ratio on chromosome 7q32.3-qter: a genome-wide scan in the Framingham study[END_REF]. In summary, several common polymorphisms of the TPK1 gene appear to be associated with birth weight. It cannot be formally excluded that the currently associated SNPs are only markers of neighbouring unidentified functional SNPs that are responsible for the observed association. Also, our study is underpowered to identify associations with polymorphisms that are rare or exert only modest effects. Finally, the fact that we were not able to reconstruct all haplotypes in all families reduced the size of the haplotype groups and thus the level of significance of our observations. Confirmation of the current association and new physiological insights are therefore necessary before genetic relationships between thiamine metabolism and human fetal growth can be established. Since thiamine deficiency remains an important public health issue in some populations, the study of thiamine status during the third trimester of pregnancy may help to investigate the postulated effects of the TPK1 genotype in these populations.

TPK1

  Variants are Associated with Birth Weight population stratification. Sex was used as a covariate in each analysis of birth weight. The null model was "Means = Mu+ sex+ß b " and the full model was "Means = Mu+sex+ ß b + ß w ". We used Bonferroni's correction for multiple tests.

  and P. Bougneres rs228584 and birth weight could be partly explained by the maternal genotype at this locus.

Table 1

 1 Primers used to amplify the TPK1 coding region

	Exon	Primer forward	Primer reverse

Table 2

 2 Details of the 43 SNPs that were genotyped in the studied cohort

				valently attached oligonucleotide probes, assembled into
				etched fiber optic bundles. Array bundles are fabricated
	SNPs	Position	Frequencies	into matrices whose configuration matches the wells of a
	rs886469	143589702	C = 0.325	standard 96-well microtiter plate. The average genotyp-
	rs726068	143591093	A = 0.325	ing error rate estimated through blind duplicated sam-
	rs2158370 rs1007808	143592701 143596709	A = 0.203 A = 0.194	pling was less than 0.01%.
	rs757387	143597931	G = 0.208
	rs994418	143601739	T = 0.333
	rs2189529	143603643	A = 0.208
	rs969356	143610971	G = 0.333
	rs41238	143624003	A = 0.292
	rs41242	143634663	G = 0.292
	rs1860265	143652570	T = 0.3
	rs962779	143657052	C = 0.281
	rs2057868	143662292	A = 0.208
	rs2189909	143695944	G = 0.067
	rs717885	143710050	A = 0.092
	rs964543	143722498	G = 0.05
	rs1476625	143738373	T = 0.017
	rs2040898	143759559	C = 0.15
	rs721492	143771803	T = 0.25
	rs990282	143774546	T = 0.373
	rs765811	143806013	A = 0.308
	rs2189738	143835631	G = 0.05
	rs228577	143838310	T = 0.449
	rs228578	143846081	T = 0.275
	rs228580	143848813	T = 0.467
	rs228581	143849129	T = 0.342
	rs228582	143857525	G = 0.467
	rs228584	143861007	T = 0.475
	rs228587	143867027	T = 0.417
	rs2371641	143870995	C = 0.407
	rs2056811	143892673	T = 0.43
	rs1320822	143900057	C = 0.424
	rs1731117	143903560	A = 0.492
	rs1635349	143909046	C = 0.267
	rs1635352	143913243	A = 0.492
	rs1635354	143916310	T = 0.225
	rs1731119	143916417	C = 0.467
	rs1635357	143918197	T = 0.467
	rs1635359	143918736	A = 0.467
	rs1629687	143924149	A = 0.466
	rs1635363	143926945	C = 0.467
	rs1618538	143929388	C = 0.457
	rs1635329	143939212	T = 0.225
	rs2371642	143951197	-

methodology and universal-primer PCR amplification reactions. SNP interrogation occurs before amplification using genomic DNA as template, and therefore specificity is not hybridization-based. The Bead Array Technology involves quantitatively pooled libraries of 3-micron beads, each containing multiple copies of co-

Table 3

 3 QTDT analysis of 43 SNPs located in the TPK1 gene region in 220 Caucasian families

	Marqueurs	p-value	p-value corrected *
	rs964543	0.008	0.34
	rs765811	0.004	0.17
	rs228577	0.003	0.13
	rs228581	0.0007	0.03
	rs228582	0.001	0.04
	rs228584	0.0007	0.03
	rs228587	0.002	0.09
	rs2371641	0.002	0.09

* For multiple tests.

Acknowledgements

D. Fradin is supported by a doctoral fellowship grant from Ministère de la Recherche, then the Alfred Jost Research Fellowship sponsored by the Serono company. We thank Mark Lathrop, Simon Heath and collaborators at the Centre National de Génotypage for genotyping HapMap SNPs. We thank an anonymous reviewer for improving the interpretation of our data.

72bp 70bp 73bp 96bp 147bp 112bp 1714bp rs886469 rs726068 rs2158370 rs1007808 rs757387 rs994418 rs2189529 rs969356 rs41238 rs41242 rs1860265 rs962779 rs2057868 rs2189909 rs717885 rs964543 rs1476625 rs2040898 rs721492 rs990282 rs765811 rs2189738 rs228577 rs228578 rs228580 rs228581 rs228582 rs228584 rs228587 rs2371641 rs2056811 rs1320822 rs1731117 rs1635349 rs1635352 rs1635354 rs1731119 rs1635357 rs1635359 rs1629687 rs1635363 rs1618538 rs1635329 rs2371642 rs886469 rs726068 rs2158370 rs1007808 rs757387 rs994418 rs2189529 rs969356 rs41238 rs41242 rs1860265 rs962779 rs2057868 rs2189909 rs717885 rs964543 rs1476625 rs2040898 rs721492 rs990282 rs765811 rs2189738 rs228577 rs228578 rs228580 rs228581 rs228582 rs228584 rs228587 rs2371641 rs2056811 rs1320822 rs1731117 rs1635349 rs1635352 rs1635354 rs1731119 rs1635357 rs1635359 rs1629687 rs1635363 rs1618538 rs1635329 rs2371642

rs886469 rs726068 rs2158370 rs1007808 rs757387 rs994418 rs2189529 rs969356 rs41238 rs41242 rs1860265 rs962779 rs2057868 rs2189909 rs717885 rs964543 rs1476625 rs2040898 rs721492 rs990282 rs765811 rs2189738 rs228577 rs228578 rs228580 rs228581 rs228582 rs228584 rs228587 rs2371641 rs2056811 rs1320822 rs1731117 rs1635349 rs1635352 rs1635354 rs1731119 rs1635357 rs1635359 rs1629687 rs1635363 rs1618538 rs1635329 rs2371642 D' 0,0 0,2 0,4 0,6 0,8 1,0

74bp 59bp 72bp 70bp 73bp 96bp 147bp 112bp 1714bp rs886469 rs726068 rs2158370 rs1007808 rs757387 rs994418 rs2189529 rs969356 rs41238 rs41242 rs1860265 rs962779 rs2057868 rs2189909 rs717885 rs964543 rs1476625 rs2040898 rs721492 rs990282 rs765811 rs2189738 rs228577 rs228578 rs228580 rs228581 rs228582 rs228584 rs228587 rs2371641 rs2056811 rs1320822 rs1731117 rs1635349 rs1635352 rs1635354 rs1731119 rs1635357 rs1635359 rs1629687 rs1635363 rs1618538 rs1635329 rs2371642 74bp 59bp 72bp 70bp 73bp 96bp 147bp 112bp 1714bp rs886469 rs726068 rs2158370 rs1007808 rs757387 rs994418 rs2189529 rs969356 rs41238 rs41242 rs1860265 rs962779 rs2057868 rs2189909 rs717885 rs964543 rs1476625 rs2040898 rs721492 rs990282 rs765811 rs2189738 rs228577 rs228578 rs228580 rs228581 rs228582 rs228584 rs228587 rs2371641 rs2056811 rs1320822 rs1731117 rs1635349 rs1635352 rs1635354 rs1731119 rs1635357 rs1635359 rs1629687 rs1635363 rs1618538 rs1635329 rs2371642 rs886469 rs726068 rs2158370 rs1007808 rs757387 rs994418 rs2189529 rs969356 rs41238 rs41242 rs1860265 rs962779 rs2057868 rs2189909 rs717885 rs964543 rs1476625 rs2040898 rs721492 rs990282 rs765811 rs2189738 rs228577 rs228578 rs228580 rs228581 rs228582 rs228584 rs228587 rs2371641 rs2056811 rs1320822 rs1731117 rs1635349 rs1635352 rs1635354 rs1731119 rs1635357 rs1635359 rs1629687 rs1635363 rs1618538 rs1635329 rs2371642

rs886469 rs726068 rs2158370 rs1007808 rs757387 rs994418 rs2189529 rs969356 rs41238 rs41242 rs1860265 rs962779 rs2057868 rs2189909 rs717885 rs964543 rs1476625 rs2040898 rs721492 rs990282 rs765811 rs2189738 rs228577 rs228578 rs228580 rs228581 rs228582 rs228584 rs228587 rs2371641 rs2056811 rs1320822 rs1731117 rs1635349 rs1635352 rs1635354 rs1731119 rs1635357 rs1635359 rs1629687 rs1635363 rs1618538 rs1635329 rs2371642

D'

0,0 0,2 0,4 0,6 0,8 1,0

Figure 1

Localisation and linkage disequilibrium of the 43 SNPs that were genotyped in the studied cohort.