Hélène Guegan

Tijana Stajner

Branko Bobić

Cindy Press

Rares T Olariu

Kjerstie Olson

Jelena Srbljanovic

Jose G Montoya

Olgica Djurković-Djaković

Florence Robert

Tijana Stäjner

Branko Bobic

Prof Florence Robert-Gangneux
email: florence.robert-gangneux@univ-rennes1.fr

Maternal anti-Toxoplasma Treatment during Pregnancy is Associated with Reduced Sensitivity of Diagnostic Tests for Congenital Infection in the Neonate

Keywords: Congenital toxoplasmosis, Toxoplasma gondii, serology, IgM, IgA, western-blot, maternal treatment, neonatal diagnosis, qPCR

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Toxoplasmosis is a widespread protozoan foodborne infection [START_REF] Blaga | Toxoplasma gondii in beef consumed in France: regional variation in seroprevalence and parasite isolation[END_REF][START_REF] Rousseau | Assessing viability and infectivity of foodborne and waterborne stages (cysts/oocysts) of Giardia duodenalis, Cryptosporidium spp., and Toxoplasma gondii: a review of methods[END_REF], affecting about one third of humans, with great differences in prevalence according to geographical areas [START_REF] Pappas | Toxoplasmosis snapshots: global status of Toxoplasma gondii seroprevalence and implications for pregnancy and congenital toxoplasmosis[END_REF]. Hence, public health policies vary among countries, as the disease burden is diversely appreciated [START_REF] Torgerson | The global burden of congenital toxoplasmosis: a systematic review[END_REF][START_REF] Bissati | Global initiative for congenital toxoplasmosis: an observational and international comparative clinical analysis[END_REF]. Indeed, Toxoplasma gondii infection is largely asymptomatic, except in immunocompromised subjects. It can also be responsible for congenital infection when primary infection is acquired during pregnancy. The only way to know if a pregnant woman has acquired toxoplasmosis during pregnancy is to determine her serological status at the beginning of gestation, and repeat serological testing regularly until delivery, if it shows an absence of protective immunity. Such serological screening has been implemented in some countries including France, Austria, Belgium, Italy [START_REF] Robert-Gangneux | Epidemiology of and diagnostic strategies for toxoplasmosis[END_REF][START_REF] Tomasoni | Multidisciplinary approach to congenital Toxoplasma infection: an Italian nationwide survey[END_REF][START_REF] Prusa | Congenital toxoplasmosis in Austria: Prenatal screening for prevention is cost-saving[END_REF], to allow early maternal anti-Toxoplasma therapy, and reduce vertical transmission and severe fetal damages [START_REF] Syrocot ; Thiébaut | Effectiveness of prenatal treatment for congenital toxoplasmosis: a meta-analysis of individual patients' data[END_REF][START_REF] Hotop | Efficacy of rapid treatment initiation following primary Toxoplasma gondii infection during pregnancy[END_REF][START_REF] Cortina-Borja | Prenatal treatment for serious neurological sequelae of congenital toxoplasmosis: an observational prospective cohort study[END_REF][START_REF] Kieffer | Risk factors for retinochoroiditis during the first 2 years of life in infants with treated congenital toxoplasmosis[END_REF][START_REF] Wallon | Congenital toxoplasma infection: monthly prenatal screening decreases transmission rate and improves clinical outcome at age 3 years[END_REF][START_REF] Prusa | Amniocentesis for the detection of congenital toxoplasmosis: results from the nationwide Austrian prenatal screening program[END_REF][START_REF]It is not only the cat that did it: How to prevent and treat congenital toxoplasmosis[END_REF]. The usual treatment consists in spiramycin (SPI) administration until delivery [START_REF] Konstantinovic | Treatment of toxoplasmosis: Current options and future perspectives[END_REF], but if prenatal diagnosis demonstrates the presence of Toxoplasma DNA in amniotic fluid, it is strongly recommended to switch SPI to the association of pyrimethamine and sulfadiazine (PYR-SD), as the latter is more potent to reduce fetal sequelae [START_REF] Mandelbrot | Prenatal therapy with pyrimethamine + sulfadiazine vs spiramycin to reduce placental transmission of toxoplasmosis: a multicenter, randomized trial[END_REF]. However, in about 10% of cases, neonates are diagnosed after birth, despite a negative prenatal diagnosis, and in about 30% of cases, prenatal diagnosis is not performed because of late infection during gestation [START_REF] Robert-Gangneux | Clinical Relevance of Placenta Examination for the Diagnosis of Congenital Toxoplasmosis[END_REF]. Therefore, postnatal diagnosis is essential to diagnose infected neonates and start treatment, as recommended [START_REF] Moncada | Toxoplasmosis in the fetus and newborn: an update on prevalence, diagnosis and treatment[END_REF][START_REF] Maldonado | Diagnosis, Treatment, and Prevention of Congenital Toxoplasmosis in the United States[END_REF][START_REF] Peyron | Maternal and Congenital Toxoplasmosis: Diagnosis and Treatment Recommendations of a French Multidisciplinary Working Group[END_REF].

Clinical and biological work-up for postnatal diagnosis of congenital toxoplasmosis relies on a combination of methods including, parasite detection in placenta or amniotic fluid collected during delivery [START_REF] Robert-Gangneux | Clinical Relevance of Placenta Examination for the Diagnosis of Congenital Toxoplasmosis[END_REF], cord blood or newborn blood, by PCR and serological screening [START_REF] Olariu | Polymerase chain reaction in cerebrospinal fluid for the diagnosis of congenital toxoplasmosis[END_REF][START_REF] Pomares | Laboratory Diagnosis of Congenital Toxoplasmosis[END_REF]. The serological workup is now wellestablished [START_REF] Robert-Gangneux | Epidemiology of and diagnostic strategies for toxoplasmosis[END_REF][START_REF] Peyron | Maternal and Congenital Toxoplasmosis: Diagnosis and Treatment Recommendations of a French Multidisciplinary Working Group[END_REF][START_REF] Murat | Human toxoplasmosis: which biological diagnostic tests are best suited to which clinical situations?[END_REF] and usually relies on the detection of specific IgA or IgM in the neonate serum. As specific IgG antibodies developed by the mother are transferred to the fetal compartment, the only way to detect specific IgG synthesized by the neonate himself, is to characterize anti-Toxoplasma IgG profiles of mother and newborn paired serum samples at birth by western-blotting. This technique has been added to the routine serological workup since many years in most reference labs in France [START_REF] Dupont | Usefulness of Western blot in serological follow-up of newborns suspected of congenital toxoplasmosis[END_REF][START_REF] Robert-Gangneux | Performance of a Western blot assay to compare mother and newborn anti-Toxoplasma antibodies for the early neonatal diagnosis of congenital toxoplasmosis[END_REF], and is widely used across Europe. Sensitivity data of these techniques have been published over years, but show various performances according to studies and countries.

A possible impact of maternal treatment has been suggested for parasite detection in placenta or IgM detection in neonates, with a higher rate of positive qPCR or positive IgM, respectively, when mothers had received SPI, compared to PYR-SD [START_REF] Robert-Gangneux | Clinical Relevance of Placenta Examination for the Diagnosis of Congenital Toxoplasmosis[END_REF][START_REF] Bessières | Neonatal screening for congenital toxoplasmosis in a cohort of 165 women infected during pregnancy and influence of in utero treatment on the results of neonatal tests[END_REF]. However, both studies were conducted in France, where women usually receive specific therapy during pregnancy, thus the comparison of the sensitivity of biological tests in neonates born to treated and untreated mothers was not addressed. Recently, Olariu et al. [START_REF] Olariu | Congenital toxoplasmosis in the United States: clinical and serologic findings in infants born to mothers treated during pregnancy[END_REF] reported a possible impact of maternal treatment on IgA and IgM detection, but the number of treated cases was small, and the trimester of infection, which was shown to influence IgA and IgM antibody detection in the neonate, was not taken into account.

Therefore, the aim of the present study was to assess the impact of maternal anti-Toxoplasma treatment during pregnancy on the sensitivity of various tests in neonates at birth, using a multivariate analysis performed on data from three countries with different national policies, i.e. maternal screening and treatment (France), occasional maternal screening (Serbia), and no maternal screening (USA).

MATERIALS AND METHODS

Ethics

All analysis were performed during routine work-up, as implemented in the three participating centers.

Data were recorded anonymously. The study design was approved by the local ethics committee of the University Hospital of Rennes (approval number: #20.08).

Patients

All congenitally infected infants diagnosed over a 10-year period (2010-2019) in the three labs (Dr. Jack S. Other relevant data were recorded for analysis: age of gestation at the time of maternal infection, type, date of onset and duration of anti-Toxoplasma targeted therapy, gender of the newborn, clinical signs at birth or during the first year of life. When the date of maternal infection could not be known with accuracy, due to the absence of early serological screening, the estimation was graduated as "1 st , 2 nd or 3 rd trimester", according to clinical or imaging findings or serological results during pregnancy or "undetermined" when discovered only after birth.

Remington

Serological methods

Anti-Toxoplasma IgM were detected by ELISA (Platelia Toxo IgM, Biorad, Marnes-la-Coquette, France) and ISAGA (BioMérieux, Marcy-l'Etoile, France) in CNLTS and RUH-PL, and only by ISAGA in JSRLSD. Anti-Toxoplasma IgA were detected by ISAGA IgA (BioMérieux) in CNLTS and RUH-PL, and by an in-house ELISA in JSRLSD. Comparison of maternal and neonatal IgG and IgM antibody profiles was performed using the Toxoplasma WB IgG/IgM assay (LDBIO, Lyon, France) in CNLTS and RUH-PL.

ELISA indexes for IgM and IgA were positive when ≥1 and negative when <0.8. For performance calculation, ELISA indexes in grey zone were merged with positive results. ISAGA IgM and IgA scores were positive when ≥3, according to the manufacturer's recommendations. When a positive result was obtained on a cord blood sample using a quantitative test, it was taken into account only if it could be confirmed on peripheral blood or if WB could rule out contamination of cord blood by maternal IgM (different profiles).

WB profiles of the mother and her baby were compared to determine if additional bands were observed in the neonatal serum (neosynthesis by the infected neonate). When cord blood samples were tested, if the IgM patterns of the mother and baby were similar, then the cord blood was considered contaminated by the maternal serum and was excluded from the study. When maternal and neonatal patterns differed by only one faint band, the WB was considered as doubtful. In both instances, if there were no other arguments in favor of the diagnosis of infection, all tests were repeated on a neonatal blood sample collected several days later.

Toxoplasma real-time PCR

For parasite DNA detection, 200 µL of total blood or cerebrospinal fluid (CSF) were extracted using Qiamp DNA Mini-kit (Qiagen) and 5 µL of DNA was used for amplification. In CNLTS and RUH-PL, the Toxoplasmaspecific quantitative real-time PCR targeted the repetitive rep529 sequence and was carried out as previously described [START_REF] Robert-Gangneux | Clinical Relevance of Placenta Examination for the Diagnosis of Congenital Toxoplasmosis[END_REF][START_REF] Pomares | Is Real-Time PCR Targeting Rep 529 Suitable for Diagnosis of Toxoplasmosis in Patients Infected with Non-Type II Strains in North America?[END_REF][START_REF] Stajner | Prenatal and Early Postnatal Diagnosis of Congenital Toxoplasmosis in a Setting With No Systematic Screening in Pregnancy[END_REF]. In JSRLSD, the PCR used two target sequences from the B1 gene [START_REF] Olariu | Polymerase chain reaction in cerebrospinal fluid for the diagnosis of congenital toxoplasmosis[END_REF].

Statistical analysis

All serological and PCR results were included as dichotomous variables (positive / negative), with borderline test results classified as positive, and were analyzed according to maternal treatment.

An univariate analysis was performed on population characteristics according to treatment group. The association of the different test results with qualitative variables (gender, type of sample, trimester of pregnancy, treatment group) was analyzed using Pearson's χ2 test or Fisher's exact test. For continuous variables (age at time of sampling and duration of treatment) for which the Skewness and Kurtosis tests showed non-Gaussian distribution, we used the nonparametric Mann-Whitney U test, and the distributions were displayed as medians with interquartile ranges [25th;75th percentile].

Multivariate analysis was then carried out, and a logistic regression model was set up with each particular test result as an outcome variable and the trimester of infection, the gender, the newborn age at sampling (in days), the center, and the maternal treatment (yes / no) as covariates. The duration of treatment (by categories ≤8 / >8 weeks or in number of weeks) and the type of treatment (SPI or SPI ± PYR-SD) were further analyzed on the treated group. Only variables that were found to be significant by univariate analysis, with P < 0.2, were tested forward by stepwise Wald logistic regression. The Odds ratio ([OR]) and 95% confidence intervals (95% CI) were used to describe the associated factors if P < 0.05.

All statistical tests and procedures were performed using the IBM SPSS 21 statistical package (IBM, NY, USA).

RESULTS

Patient file

A total of 115 mother-neonate pairs were included from the 3 centers; 46 from France, 26 from Serbia and 43 from the USA. Congenital infection was diagnosed antenatally (n=27) and/or after birth on the basis of neosynthesized antibody detection or positive PCR on newborn samples (Supplementary Table 1). The first positive post-natal test was obtained as soon as birth until one year of age when treatment was stopped (Supplementary Table 1). Neonates were aged from 0 to 35 days (median days 3 [1; 11], when the first blood sample was obtained and included in the study. The time of sampling was earlier in newborns from treated mothers than from untreated ones, as a result of the follow-up protocol implemented in case of maternal Toxoplasma infection in Europe (Table 1). Blood samples included 46 cord blood and 69 peripheral blood samples. Most samples were taken before 5 days of life (n=70) (Table 1). The frequency of detection of specific IgG, IgM, or IgA neosynthesized antibodies did not depend on the sample type in any of the analyzed diagnostic tests (p>0.05). Maternal seroconversion occurred during the first (T1), second (T2) and third (T3) trimester in 14 (12.2%), 28 (24.3%) and 36 (31.3%) of cases, respectively, while the date of maternal infection remained unknown in 37 (32.2%), of which 34 were symptomatic cases (Table 1).

An anti-Toxoplasma-targeted therapy during pregnancy was given to 61 (53.0%) pregnant women. The proportion of treated women did not differ according to the trimester of seroconversion (Table 1). As expected, the median duration of treatment was significantly shorter when infection was diagnosed in T3 (median weeks 5 [2; 7]) in comparison with T2 (median weeks 12 [7.25; 14]) and T1 (median weeks 17.5 [8; 24], p<0.001), respectively. The treatment consisted of SPI alone (n=34), PYR-SD (n=8) or SPI followed by PYR-SD (n=19). Interestingly, the proportion of infections acquired during the first, second and third trimester didn't differ among treated and untreated mothers (Table 1), ruling out a possible interplay of these two parameters in the interpretation of results.

The incidence of Toxoplasma-related clinical manifestations within the first year of follow-up reached 53.1% (60/113) (Table 1). The main disorders included cerebral lesions, i.e intracranial calcifications, ventriculomegaly and meningitis in 47/60 (78.3%), ocular lesions in 26/60 (43.3%), and growth retardation/prematurity in 7/60 (11.7%) neonates. The rate of newborns with clinical signs was significantly lower when treatment was given during pregnancy (31.1% versus 75.9% in treated and untreated groups, respectively) (Table 1).

Maternal treatment is associated with lower frequency of detection of anti-Toxoplasma IgM and IgG in the neonate, but not of IgA

As a first approach to investigate the impact of maternal anti-Toxoplasma treatment on the results of diagnostic methods, we calculated the overall sensitivity of each test for prenatally treated and untreated infants. The overall sensitivity of IgM detection by ELISA and/or ISAGA was lower in infants born to treated mothers (p<0.0001, Table 2). However, when analyzing separately each technique, the ELISA-based IgM detection was not affected by treatment, whereas the sensitivity of IgM ISAGA dropped from 90.7% to 53.3%, when treatment was administered (p<0.0001, Table 2). WB-based IgM detection was also significantly lower in the treated group, as it was positive in only 55.8%, compared to 88.2% without treatment (p<0.05). This reduction was also observed when gathering doubtful IgM WB profiles (only one additional faint band obtained with the neonate's profile, compared to the mother's IgM profile) with negative results instead of with the positive ones (p=0.009, data not shown). An above two-fold reduction of neosynthesized IgG detection by WB was also observed in neonates from treated mothers, 82.3% versus 37.7% (p<0.01). No significant differences were observed in the detection of specific IgA (ELISA or ISAGA) according to maternal treatment (Table 2).

To further analyze the differences observed between ISAGA and ELISA for IgM detection, we evaluated the sensitivity on serum samples analyzed concomitantly with both techniques (Serbian and French neonates). Results showed that poorer ISAGA sensitivity was associated with maternal treatment, whereas ELISA sensitivity was even in both treated and untreated groups (Table 3).

As the absence of specific IgM detection in the neonate could be solely due the physiologic disappearance of antibodies before birth when infection occurred in early pregnancy, we then performed a multivariate analysis to confirm the results obtained by univariate analysis. Interestingly, results showed that treatment significantly altered IgM detection by ISAGA and WB (p<0.001 and p<0.01, respectively), as well as neosynthetized IgG detection by WB (Table 4), thus confirming univariate analysis. Multivariate analysis also showed that the trimester of maternal infection influenced the positivity of these serological tests with more frequent positivity for infections acquired during the third trimester, while the age of the newborn at the time of sampling had no effect (Table 4 andSupplementary Table 2). Unexpectedly, the gender was significantly associated with IgM detection, with male and female infants being most likely to have positive ISAGA IgM, and positive WB IgM, respectively (Table 4 and Supplementary Table 2).

The duration of anti-Toxoplasma therapy during pregnancy is unequally associated with reduced sensitivity of serological tests

To further analyze if the duration of maternal anti-Toxoplasma therapy influenced the sensitivity of diagnostic methods, we divided the group of 61 infants from treated mothers into two sub-groups, according to the length of maternal treatment, i.e. ≤8 weeks or >8 weeks of any treatment. Interestingly, the duration of treatment interfered heterogeneously with the detection of IgM, IgA or IgG. The detection of specific IgA, regardless of the method, was not altered, but when distinguishing ISAGA and ELISA, it appeared that ISAGA sensitivity was significantly reduced if treatment course was >8 weeks (p <0.05, Table 5). Sensitivity of IgM detection, whatever the technique used (ELISA, ISAGA or WB), was dramatically decreased when treatment exceeded 8 weeks (p<0.001, p<0.01, p=0.001, respectively) (Table 5). Overall, the range of IgM detection decreased from 72-79% to 29-32%, when treatment lasted ≤8 weeks and >8 weeks, respectively, depending on the method. By contrast, IgG detection by WB was not significantly modified according to the cut-off of 8 weeks of treatment (Table 5). The duration of treatment seems to have a progressive effect, as illustrated in Fig 1 .    Multivariate analysis using the same cut-off of treatment duration (≤8 weeks or >8 weeks), confirmed these findings (Table 5). Additionally, when taking into account the precise duration of treatment (number of weeks), a longer duration of treatment was associated with a lower sensitivity of ISAGA IgA, as well as of IgM detection by any technique, but no impact on neosynthesized IgG detection by WB (Table 6).

Overall, among the 61 treated neonates, no statistically significant difference (p > 0.05) was found between neonates treated with SPI or PIR-SD ± SPI, in any serological test results (Table 6).

Prenatal therapy is associated with decreased sensitivity of the detection of parasite DNA in the neonate at birth Eighty-four neonate specimens consisting of blood (n=66) or CSF samples (n=18) were also collected for Toxoplasma detection by qPCR. The sensitivity dropped from 43.6% to 24.4% in neonates from untreated and treated mothers, respectively (Table 2). When the precise duration of treatment was analyzed for each type of treatment, regression analysis showed that there was no effect of SPI alone, whereas there was a significant time-dependent effect of PYR-SD on the DNA detection by PCR (Table 6).

DISCUSSION

The laboratory diagnosis of congenital toxoplasmosis at birth requires the use of multiple techniques to prove infection of the neonate, through detecting either antibody neosynthesis by the newborn, or circulating parasite DNA. Postnatal screening of neonates is paramount in three situations: i) when evocative clinical signs are present at birth and the mother has not benefited from Toxoplasma serological screening during gestation, ii) when the mother acquired toxoplasmosis during gestation and the prenatal diagnosis was negative, and iii) when the mother acquired toxoplasmosis, but prenatal diagnosis was not performed (late infection during pregnancy).

As most French women with primary infection during pregnancy are given specific treatment, we included data of infants from Serbia, where maternal screening has been only recently adopted, and from the USA, where women are diagnosed and treated only in case of ultrasound anomalies. We chose to include only early samples, because we wanted to draw conclusions on the effect of maternal on the early diagnosis of congenital toxoplasmosis. To ensure perfect interpretation of IgM or IgA detection in cord blood samples, we considered them as positive only if IgM neosynthesis by the newborn was confirmed by WB, thus ruling out maternal contamination for both IgM and IgA.

Using univariate and multivariate analyses, we found strong evidence that maternal treatment was associated with a dramatic decrease in the sensitivity of IgM and neosynthesized IgG in the neonate.

Whereas a direct impact of treatment on parasite replication and dissemination was expected, as it is assumed to reduce parasite loads [START_REF] Bessières | Neonatal screening for congenital toxoplasmosis in a cohort of 165 women infected during pregnancy and influence of in utero treatment on the results of neonatal tests[END_REF], the demonstration of an impact on antibody synthesis by the newborn is a novelty. Indeed, it could be hypothesized that the absence of IgM or IgA detection in newborns was due to a transient synthesis by the fetus, particularly if infection occurred early in gestation. Indeed, we observed that the positivity of tests was associated with the trimester of infection (Table 4), with a higher detection rate for infections acquired during the third trimester (Suppl. Table 2), which also coincides with shorter courses of treatment. However, there was no differences in the time of infection among treated and untreated mothers (Table 1), thus the time of infection cannot bias the conclusions on the effect of treatment. Taken together, multivariate analysis clearly demonstrated an association of treatment with a lower sensitivity of IgM and neosynthesized IgG antibody detection that could be explained by a delayed synthesis. Additionally, this effect was time-dependent, with an apparent cut-off of 8 weeks of any treatment (SPI or PYR-SD) for most serological tests. Besides, the ISAGA technique was more impacted by maternal treatment than the ELISA technique. Since the IgM ISAGA technique uses "global" antigens from entire parasites, this finding would suggest that maternal treatment could more likely affect the ability of the fetus to react against surface antigens, than against cytoplasmic antigens.

Why SPI and PYR-SD administered to the mother had roughly the same effect on the fetus or neonate antibody response is difficult to explain, as SPI is known as hardly transferring through the placental barrier. A previous study by Gilbert et al. found no impact of maternal treatment on the sensitivity of IgM detection by multivariate analysis, but the group treated with PYR-SD was compared to a control group including both SPI-treated and untreated mothers [START_REF] Gilbert | Screening for congenital toxoplasmosis: accuracy of immunoglobulin M and immunoglobulin A tests after birth[END_REF].

In contrast to our results, Naessens et al. investigated IgM detection in peripheral blood samples from 86 congenitally infected newborns and found no significant effect of treatment despite an apparent decrease of sensitivity (85% in untreated versus 25% in treated, p=0.19) ; they attributed it to the gestational age of maternal infection rather than to prenatal treatment [START_REF] Naessens | Diagnosis of congenital toxoplasmosis in the neonatal period: A multicenter evaluation[END_REF]. However, the vast majority of mothers (75%) had been treated, and the duration of treatment was not known with accuracy, nor the time of maternal infection, as monthly serological screening was not current practice in most centers. In a Brazilian study primarily aiming at measuring the duration of IgM synthesis by infected infants, Lago et al. observed that in 23/28 neonates whose mother had been diagnosed with seroconversion during pregnancy, IgM were detected during the first month of life. However, the difference in sensitivity between treated (75%) and untreated (88%) groups was not statistically significant (p=0.4) [START_REF] Lago | Presence and duration of anti-Toxoplasma gondii immunoglobulin M in infants with congenital toxoplasmosis[END_REF].

While IgA detection appeared not to be affected by treatment using univariate analysis, there was a trend towards a time-dependent effect of treatment in infants from treated mothers, irrespective of the type of treatment. Again, this was significant only with the ISAGA technique, but it must be acknowledged that there was a small number of samples analyzed with ELISA. Interestingly, it has been shown in experimental murine infection that specific IgA synthesis was the least affected of all antibody classes by several anti-Toxoplasma chemotherapeutic regimens [START_REF] Alvarado-Esquivel | Antiparasitic treatment suppresses production and avidity of Toxoplasma gondii-specific antibodies in a murine model of acute infection*[END_REF].

In contrast, despite a significant effect of treatment on the detection of neosynthesized IgG by WB, it was not related to the duration of maternal treatment. It was not related to the time of sampling, i.e. the age of the newborn, either (Table 4). In infants from untreated mothers, neosynthesized IgG were found in 4/5 (80%) in the early days after birth (0-4 days) and in 7/9 (78%) from day 15 to 35 (data not shown). Few reports are available on the detection of neosynthesized IgG using WB. In a cohort of 55 neonates, a French multicenter study reported a sensitivity of IgG WB similar to ours at birth (37.7% in the treated group), increasing to 40% during the first 10 days of life, and to 63.3% between 0.5 and 1.5 months [START_REF] Pinon | Strategy for diagnosis of congenital toxoplasmosis: evaluation of methods comparing mothers and newborns and standard methods for postnatal detection of immunoglobulin G, M, and A antibodies[END_REF].

Similar findings were published in Brazil, where the IgG WB was positive in 40% (6/15) of infected newborns born to treated mothers up to 3 months of age [START_REF] Capobiango | Evaluation of the Western blotting method for the diagnosis of congenital toxoplasmosis[END_REF]. In another Brazilian series of newborns aged up to 1 month and born to women who had not been given treatment during pregnancy, the sensitivity of IgG WB reached 73.5 % (130/177), which is not far from the results obtained in our untreated group (82.3%) [START_REF] Machado | IgG and IgM western blot assay for diagnosis of congenital toxoplasmosis[END_REF]. However, they found a lower sensitivity of IgM WB (54.8 %) compared to that of IgG WB.

In a previous study, Olariu et al (2019) analyzed IgM and IgA detection in infants from 25 treated and 164 untreated mothers, and similarly found that IgM were less frequently detected in infants of treated mothers (44%) than of untreated mothers (86.6%, p<0.001) [START_REF] Olariu | Congenital toxoplasmosis in the United States: clinical and serologic findings in infants born to mothers treated during pregnancy[END_REF]. The detection of IgA was not modified according to maternal treatment, either (p=0.06). However, they did not take into account the gestational age at the time of maternal infection, which is a major factor impacting the sensitivity of serological tests [START_REF] Bessières | Neonatal screening for congenital toxoplasmosis in a cohort of 165 women infected during pregnancy and influence of in utero treatment on the results of neonatal tests[END_REF][START_REF] Naessens | Diagnosis of congenital toxoplasmosis in the neonatal period: A multicenter evaluation[END_REF]. In addition, samples were obtained within the first six months of life, whereas in the present study, we focused on the results obtained early after birth (mean age of 7 days).

It is interesting to note that, in case of maternal anti-Toxoplasma treatment, the detection of IgA was the most sensitive approach to detect neonatal infection, outperforming IgM and IgG-based screening (68.4% versus 59% and 38%, respectively), while it was less sensitive in untreated cases.

Still, the overall sensitivity of IgA and IgM detection (71.8% and 73.9%, respectively) is quite high compared to previous reports, which showed sensitivity ranges from 47.8 to 72.5% and 44 to 67.5%, for IgA and IgM, respectively, irrespective of the technique [START_REF] Murat | Human toxoplasmosis: which biological diagnostic tests are best suited to which clinical situations?[END_REF][START_REF] Bessières | Neonatal screening for congenital toxoplasmosis in a cohort of 165 women infected during pregnancy and influence of in utero treatment on the results of neonatal tests[END_REF][START_REF] Gilbert | Screening for congenital toxoplasmosis: accuracy of immunoglobulin M and immunoglobulin A tests after birth[END_REF][START_REF] Pinon | Strategy for diagnosis of congenital toxoplasmosis: evaluation of methods comparing mothers and newborns and standard methods for postnatal detection of immunoglobulin G, M, and A antibodies[END_REF].

Whereas the contribution of PCR testing in AF has been extensively evidenced for prenatal diagnosis, the use of PCR on newborn blood or CSF at birth has been sparsely investigated [START_REF] Olariu | Polymerase chain reaction in cerebrospinal fluid for the diagnosis of congenital toxoplasmosis[END_REF]. In our cohort, the sensitivity of blood PCR was low (28.6%) but it is interesting to note that it ranged from 23% to 39% in treated and untreated infants, respectively. Unexpectedly, while univariate analysis was significant, multivariate analysis showed that treatment onset did not impact PCR positivity. However, among treated neonates, the longer the treatment, the lower the sensitivity of PCR, provided that the treatment was PYR-SD, which is in agreement with the expected in utero anti-parasitic effect of this bi-therapy. The trimester of pregnancy had no impact on the positivity of PCR, while it might be expected to find more positive PCR results when infection occurred at the end of pregnancy, due to a shorter course of treatment. These findings may suffer from a bias of center, as the proportion of positive cases was higher in the JSRLSD (38% versus 23% in RUHPL), but 37 cases were excluded from the multivariate analysis because of missing data, thus the power of the analysis may have been impaired.

The strength of our study is that it was performed on a well-balanced cohort of prenatally-treated and untreated infants, diagnosed prospectively in three reference centers. A limitation can result from the uncertainty of the time of maternal infection in some US cases, as no serological follow-up was undertaken, which led us to discard 32% of cases for the multivariate analysis.

Taken together, we provide evidence that anti-Toxoplasma therapy in the mother may contribute to set back serological confirmation in the neonate, and underlines the need for careful serological follow-up of neonates even if the workup at birth is normal. We also recall that prenatal diagnosis can detect at most 90% of infected fetus, thus postnatal serological follow-up is essential.

ACKNOWLEDGMENTS

LEGENDS TO FIGURES

  Laboratory for Specialty Diagnostics, Palo Alto, CA, USA, (JSRLSD), National Reference Laboratory for toxoplasmosis from Serbia (CNLTS) and Rennes University Hospital Parasitology lab (RUH-PL)) were retrospectively included if a blood sample had been analyzed at birth or during the first month of life. Diagnosis of congenital toxoplasmosis relied on a positive prenatal diagnosis (parasite DNA detection by qPCR on amniotic fluid), and/or detection of specific IgM or IgA in peripheral blood at >7 days of life, or positive Toxoplasma qPCR in peripheral blood or cerebrospinal fluid, or detection of neosynthesized IgG or IgM by western-blot.

  This study was supported by the Ministry of Education, Science and Technological Development of Serbia, through project no. III 41019; and by a grant to the Institute for Medical Research (contract no. 451-03-68/2020-14/200015). The authors also received a grant from Campus France (PHC Pavle Savic no.40734WC) (Serbian project number 451-03-01963/2017-09/15).

Fig. 1 :

 1 Fig. 1: Sensitivity of diagnostic tests according to the duration of treatment.

* 29 Figure 1 E

 291 Figure 1

Table 1 .

 1 Population characteristics (N=115) and comparison according to treatment group 464

	465
	Characteristics	Maternal treatment during	Univariate
		pregnancy	analysis $
		No (N = 54)	Yes (N = 61)	p
	Sex ratio (M/F)	0.69 (22/32)	1.10 (32/29)	0.209, ns
	Type of blood sample, n (%)			<0.0001
	Cord blood (n=46)	4 (7.4)	42 (68.8)
	Peripheral blood (n=69)	50 (92.6)	19 (31.1)
	Time of blood sampling, median	10 (4-17)	1 (0-3)	<0.0001
	(interquartile range)
	0-4 days of life (n=70), n (%)	16 (29.6)	54 (88.6)
	5-14 days of life (n=21), n (%)	20 (37.0)	1 (1.6)
	15-35 days of life (n=24), n (%)	18 (33.3)	6 (9.8)
	Trimester of maternal infection, n (%)			0.063, ns
	TI (n=14)	4 (7.4)	10 (16.4)
	TII (n=28)	2 (3.7)	26 (42.6)
	TIII (n=36)	11 (20.4)	25 (41.0)
	ND (n=37)	37 (68.5)	0
	Clinical manifestation during the first			<0.0001
	year of life, n (%)
	Yes (n=60)	41 (75.9)	19 (31.1)
	No (n=53)	11 (20.4)	42 (68.9)
	ND (n=2)	2 (3.7)	0
	Type of treatment, n (%)	na		na
	SPI alone		34 (55.7)
	SPI then PYR-SD		19 (31.1)
	PYR-SD alone		8 (13.1)
	467
	468
	469

$ Pearson's 2 test or + Fisher's exact test compared distributions in treated and untreated groups 466 na: not applicable; nd: not determined; ns: not significant; SPI: spiramycin; PYR-SD: pyrimethamine-sulfadiazine

Table 3 .

 3 Compared sensitivity of ISAGA and ELISA assays for IgM detection (N=70) according to treatment group

	476
	477
	478
	479
	480
	481

Table 4 .

 4 Multivariate analysis of the effect of treatment, trimester of maternal infection, and age at blood sampling on the sensitivity of diagnostic tests

	Outcome
	Variable

*blood or CSF ND; not determined because the p value in univariate analysis was <0.2 (results not shown)

Table 5 .

 5 Sensitivity of serological tests and qPCR in newborns from treated mothers (N = 61), and comparison according to treatment duration ≤8 or >8 weeks

	Diagnostic test	Sensitivity according to the	Univariate	Multivariate analysis §
		duration of treatment	analysis $
		% (n/N)
		≤ 8 wks	> 8 wks	p	p	OR [95% CI]
		N = 33	N = 28
	IgA (ELISA or ISAGA) (N=57)	78.1 (25/32)	56.0 (14/25)	0.091	0.079	na
	IgA (ELISA) (N=6)	33.3 (1/3)	100 (3/3)	1	nd	na
	IgA (ISAGA) (N=51)	82.8 (24/29)	50.0 (11/22)	0.017	0.009	0.161 [0.041-0.629]
	IgM (ELISA or ISAGA) (N=61)	75.6 (25/33)	39.3 (11/28)	0.005	0.04	0.208 [0.070-0.619]
	IgM (ELISA) (N=55)	76.7 (23/30)	32.0 (8/25)	0.001	0.048	0.288 [0.083-0.972]
	IgM (ISAGA) (N=60)	71.9 (23/32)	32.1 (9/28)	0.004	0.035	0.117 [0.026-0.517]
	IgM (WB) (N=52)	78.6 (22/28)	29.2 (7/24)	0.001	0.001	0.112 [0.032-0.396]
	IgG (WB) (N=53)	46.4 (13/28)	28.0 (7/25)	0.256	nd	na
	qPCR # (N=45)	30.8 (8/26)	15.8 (3/19)	0.309	nd	na
	na: not applicable
	nd: not determined because the p value in univariate analysis was <0.2

$ Fischer's exact test § multivariate analysis was done using the treatment duration as qualitative variable (>8 or ≤8 weeks); data for other covariates are not shown

Table 6 :

 6 Sensitivity of diagnostic tests according to treatment duration (weeks)

	Diagnostic		Statistical significance of treatment duration in weeks on diagnostic tests*
	test
		Any treatment (N=61)		SPI (N = 34)	PYR-SD ± SPI (N =27)
		p-value §	Regression Analysis #	p-value §	Regression analysis	p-value §	Regression analysis
		(Mann-	P	(Mann-	p	(Mann-	p
		Whitney	OR [95% CI]	Whitney	OR [95% CI]	Whitney	OR [95% CI]
		test)		test)		test)
	IgA (ISAGA)	0.052	0.047	0.069	0.056	0.375	ND
			0.929 [0.864-0.999]
	IgM (ELISA)	0.000	0.001	0.052	0.052	0.346	ND
			0.518 [0.581-0.962]
	IgM (ISAGA)	0.001	0.003	0.030	0.040	0.002	0.004
			0.815 [0.440-0.956]		0.859 [0.7430.993]		0.864 [0.778-0.953]
	IgM (WB)	0.000	0.001	0.000	0.027	0.000	0.023
			0.857 [0.761-0.965]		0.872 [0.758 -0.901]		0.845 [0.730-0.977]
	IgG (WB)	0.149	0.155	0.435	ND	0.755	ND
	qPCR	0.124	0.193	0.615	ND	0.065	0.042
							0.102 [0.011-0.902]


Jonas Feldt

Roland Assaraf
email: assaraf@lct.jussieu.fr

Subsampling the core region, towards efficient all-electron Monte Carlo calculations in molecules

Subsampling the core region, towards efficient all-electron Monte Carlo calculations in molecules

Jonas Feldt, Roland Assaraf

Introduction

Quantum Monte Carlo (QMC) methods use a stochastic approach to solve the Schrödinger equation. Their scaling with the system size considering the computational costs is very much favourable compared to deterministic quantum chemistry methods. A great flexibility in the choice of the wave function allows to efficiently treat both dynamical as well as statical correlation. As such QMC has been used extensively for the description of excited states [START_REF] Feldt | Excited-state calculations with quantum Monte Carlo[END_REF] and materials.

A great challenge for QMC methods is the unfavourable scaling with the atomic number Z. The presence of the core electrons which are at a distance O(1/Z) from the nucleus have two undesirable consequences. First, they are usually slowing down the dynamics of the valence degrees of freedom. Second, the core electrons contribute to a large degree to the energy (O(Z 2 ) for a hydrogenoid atom) and as we will emphasize later also to the statistical fluctuations. A common way to circumvent this problem is employing effective core potentials (ECP) which are simply removing the core electrons and consequently allow for efficient sampling of the remaining valence electrons. While this is a practical approach it is as well an approximate one which spoils the high accuracy which is expected from QMC. For instance, the widely used Burkatzki-Filippi-Dolg pseudopotentials 2 have been parametrized for Hartree-Fock completely disregarding the correlation energy and the error introduced by such empirical ECPs cannot be directly judged. Comparing all-electron and valence-only calculations it has been shown that the effect is even larger for properties of excited states than for the ground state. [START_REF] Scemama | Influence of pseudopotentials on excitation energies from selected configuration interaction and diffusion Monte Carlo[END_REF] So far progresses in all-electron calculations have been focused only on alleviating the first undesirable consequence of the presence of core electrons, improving the sampling, both the ergodicity and the numerical cost. For example a spatial discretization using a double grid method allows adapting the moves close to nuclei in the Diffusion Monte Carlo method (DMC), [START_REF] Nakano | Speeding up ab initio diffusion Monte Carlo simulations by a smart lattice regularization[END_REF] leading to a gain in correlation time up to a factor 10 (for Z = 118).

In the Variational Monte Carlo (VMC) framework the correlation time for large Z atoms can be reduced using spherical coordinates. [START_REF] Umrigar | Accelerated Metropolis method[END_REF] Minimizing the computational time is also a possible route, for example the two-level Metropolis sampling [START_REF] Dewing | Improved efficiency with variational Monte Carlo using two level sampling[END_REF] factorizes the wave function in two terms and lowers the number of evaluations of one of the terms. The scaling of computing multi-determinant expansion and optimizing them has been strongly reduced. [START_REF] Clark | Computing the energy of a water molecule using multideterminants: A simple, efficient algorithm[END_REF][START_REF] Filippi | Simple formalism for efficient derivatives and multideterminant expansions in quantum Monte Carlo[END_REF][START_REF] Assaraf | Optimizing the Energy with Quantum Monte Carlo: A Lower Numerical Scaling for Jastrow-Slater Expansions[END_REF] This method is based on the determinant Lemma which allows to update a Slater determinant efficiently when only a few columns are modified.

Here we focus on the second undesirable consequence due to the presence of the core electrons, namely their large contribution to the variance. We will describe a method to build an improved estimator which enables to almost completely remove the variance coming from the core electrons. The idea is to exploit that most of the fluctuations come from a few electrons in the core regions and that the core regions are physically separated. This allows to draw an improved estimator of any random variable (here the local energy) depending on the electron coordinates (a walker) based on subsampling the core region with sidewalks.

Because these sidewalks displace a few electrons at a time their numerical cost is negligible when the number of electrons N is large, in particular for a large molecule.

Theory: Subsampling

We want to compute the expectation value of a random variable X on a density π, E(X).

In the Variational Monte Carlo framework π = Ψ 2 is the square of the trial or variational wave function and X can be the local energy for the Schrödinger Hamiltonian H,

X = E L = HΨ Ψ . (1)
We suppose that there exists a small region Ω of the probability space which is responsible for most of the variance. This region will correspond here to free moves in the core region of an atom with frozen valence electrons. We define in this work a core region as the largest sphere centered on a nucleus which contains n core electrons of a given configuration. The radius of this sphere is the distance of the first valence electron to the nucleus. Note that Ω is a random subspace as it depends on the coordinates v of the valence electrons. We will first consider the conditional expectation value E(X|Ω) as an estimator. We recall the meaning of this standard notation in probability theory using the language of integrals. For a given set of valence positions v, the coordinates c of the core electrons which are subject to a constraint c ∈ C(v) which is to be closer to the nucleus than the valence electrons. For a given valence configuration v the conditional expectation value is a number

E(X|Ω(v)) = c∈C(v) X(c, v)π(c, v)dc c∈C(v) π(c, v)dc (2)
which can be interpreted numerically as a partial average of X on a subset of walkers sampling π = Ψ 2 sharing the same valence configuration v. According to the law of total expectation the random variable E(X|Ω) which depends on v is an unbiased estimator of X i.e. E(E(X|Ω)) = E(X). This standard property in probability theory can be understood also in an integral calculus formulation

E(X) = dv c∈C(v) π(c, v)X(c, v)dc = dv c∈C(v) π(c, v)E(X|Ω(v))dc (3)
The first line is a definition of the expectation value and the second line which corresponds to the definition of E(E(X|Ω)) is easy to check by replacing E(X|Ω(v)) by its expression (2).

Since the estimator E(X|Ω) depends only on the positions of valence electrons this random variable can be seen as the effective valence property which include an (exact) ECP contribution.

This estimator fluctuates much less than X because of the variance decomposition theorem (see Appendix B).

V (X) = E(V (X|Ω)) + V (E(X|Ω)), (4)
where V (X|Ω) is the conditional variance on Ω, it is the variance obtained when the valence configuration v is frozen. The expectation value E(V (X|Ω)) can be interpreted as the contribution of the core electrons to the total variance V (X). Computing E(X|Ω) is also equivalent to adding the covariate E(X|Ω) -X to X in order to cancel the effect of the core electrons on the statistical fluctuations. For a molecule (i.e. many atoms) the estimator E(X|Ω) could be applied with Ω being defined as the union of all the core regions. We prefer instead this estimator

X = X + i (E(X|Ω i ) -X) (5)
where Ω i is a set which corresponds to moving only the core electrons of the i th atom and freezing all the other electrons. The motivation is that moving a few electrons at a time will be numerically much cheaper than moving all the electrons of the core regions. However, we expect almost the same variance reduction due to the following physical consideration.

Given a valence configuration v, two distant core regions c i and c j should be close to be separable for a physical random variable X. Mathematically, denoting by c i the electronic configuration of the core region of the i th atom, the cores are separable if the core coordinates are independent for a given valence configuration v and we can write

X = α(v)+ i β i (v, c i ).
This property implies that

E(X|Ω i ) -X = E(β i |Ω i ) -β i (v, c i ). It follows that X = α(v) and of course X = E(X|Ω).
The conditional expectation values in formula (5) are not known and have to be sampled.

In practice a covariate can be constructed by carrying out M s additional steps for each of the core regions the so-called sidewalks. The main walk is carried out in the usual manner.

After each sweep of single-electron moves in the main walk, the sidewalks for the cores are started from the current configuration. After completion of the sidewalks the main walk continues from the original configuration before the sidewalks started (Figure 1). The improved estimator of X is then computed by the subsampling process i.e. using the following estimator

X(ω, M s ) = X(ω) + λ i Ms k=1 X(ω k i ) -X(ω) M s (6)
where ω represents a particular configuration in the main walk and ω k i is the configuration ω modified by k steps of the i th sidewalk (only the electrons in the i th core region differ between ω k i and ω). Regarding the local energy the control variate E L (ω k i ) -E L (ω) can be computed with a cost O(N ) for a Jastrow-Slater function (see Eq. ( 21) in appendix). This cost can be reduced to O(1) by removing terms in the expression of the control variate

E L (ω k i ) -E L (ω)
involving the far environment of the core. They are zero in the separability limit and should be small in practice. This modification is equivalent to building a generalization of Expr. ( 5) using a function parameterized by i

X = X + λ i E(X i |Ω i ) -X i (7)
In practice we use the following formula

X(ω, M s ) = X(ω) + λ i Ms k=1 X i (ω k i ) -X i (ω) M s (8)
The term multiplied by λ is of course still a control variate (its expectation value is zero).

For the local energy a simplified expression (E i L (ω i )-E i L (ω)) is presented in Eq. ( 22) when Ψ is a Jastrow-Slater function. This form has been found to be much more efficient numerically (the scaling is O(1) and the variance is about the same).

We need to optimize M s to maximize the efficiency. The efficiency of a Monte Carlo calculation is related to the time to achieve a given statistical uncertainty σ. Given a sample of size M the statistical uncertainty is σ with

σ 2 = V c M ( 9
)
where V is the variance, c is a correlation factor (c ≥ 1) which takes into account that the points in the sample are not independent. The CPU time is T = M t where t is the time for one Monte Carlo step (a sweep over the electrons). The method is all the more efficient that the cost parameter

σ 2 T = V ct (10)
is small. Note that this parameter is independent of T for a simulation sufficiently long.

Given a random variable X parametrized by M s all the parameters on the r.h.s of Eq. (10) are functions of M s , including t because the optimal parameters of the main walk may depend on X. It is natural to consider that in the limit M s = 0 (no sidewalk) we have X = X. If t s is the CPU time of the sidewalk with M s = 1 the total computational time for a general value of M s is t(M s ) + M s t s . The gain in efficiency is

G(M s ) = V (X)t(0)c(0) V ( X)(t(M s ) + M s t s )c(M s ) = 1 r(M s )(1 + M s ts t(Ms) ) c(0)t(0) c(M s )t(M s ) (11)
where r(M s ) is the reduction of variance. The parameter t s /t(M s ) should be negligible for large N , so for a large molecule M s should approach infinity and asymptotically the gain is

G ∞ = 1 r ∞ c 0 c ∞ t 0 t ∞ (12)
where r ∞ = lim Ms→∞ r(M s ) etc. . . If the three factors (ratio of variances, ratio of correlation factors, ratio of CPU times of a single step of the main walk) for large M s are transferable from an atom to a molecule the gain in numerical efficiency for large molecules can be estimated using single atoms. Next, we will consider isolated atoms before checking this transferability property.

Reduction of the Variance for single atoms

We first investigate a series of isolated atoms to understand the properties of the core subsampling as a function of Z and the number of core electrons n core . The subsampling can be done with any method involving the Metropolis scheme but with an additional rejection step when the move is leaving Ω. Such rejection does not modify the detailed balance property ensuring that π = Ψ 2 stays the invariant measure of the subsampling process.

The simulations have been carried out for the elements Li (Z = 3) to Ar (Z = 18) with a varying number of core electrons and M s = 100 • n core steps so that they are converged or very close to the converged value E(X|Ω). The limiting value can be better estimated with an hyperbolic fit as detailed in Equation 34. The results as a function of the fraction of core electrons x are shown in Figure 2. First, one can see clearly that the variance converges correctly towards 0 for an increasing number of core electrons. The zero-variance limit is obtained when all electrons are included in the subsampling and for an infinite long sidewalk.

Of course, in this limit the subsampling is exactly equivalent to the main walk itself and the computational efficiency is not improved. However, it becomes apparent that for all values of Z the two inner electrons contribute to most of the variance, from 95% for Lithium down to 50% for Argon. This trend is analyzed in more detail in Figure 3 where results are shown for n core = 2 and 10 for (Z = 11 -18) which is equivalent to the chemical core for the second period. One can see, that the gain is larger going from the smaller to the larger core definition. The parameters n core and M s should be as small as possible to obtain numerically cheap sidewalks, but they also have to be as large as possible to reduce as much as possible the variance, leading to an optimal compromise which has to be determined. In Figure 4 the convergence of V ( X)/V (X) = r(M s ) with M s is shown for a range of number of core electrons. Independently of the size of the subsystem one can distinguish an initial quick decay in the range of 0-20 steps and a 1/M S convergence to the asymptotic limit (Eq. 34). In this initial phase most of the reduction in the variance is obtained and the differently sized subsystems which look like very similar for a small number of steps increasingly separate from each other. Figure 4: The reduction of the variance V ( X)/V (X) as a function of the number of subsampling steps M s for various number of core electrons n core for a single Aluminium atom.

Transferability to molecules, gain in variance and computational time

We first check for systems of many atoms that the gain in the variance for M s = ∞ for a single atom is transferred to molecular systems and materials. We are using linear alkanes C n H 2n+2 of increasing length with n = 1 -35 and an increasingly larger part of a silicon unit cell (Fd3m) 10 with 1-8 atoms. The results for the alkanes are shown in Figure 5 in comparison to the gain for a single carbon atom represented by the dashed line. The gain for converged sidewalks, i.e. with large M S shown in blue, does not change with the length of the alkane chain. Furthermore, one can see that the gain differs by only 5%-10% from the single carbon atom case. This result confirms the transferability of the gain in variance from an atom to a molecule. It is even systematically slightly better (by 5%-10%) for the molecule, this suggests that the separability between the core and the valence regions is enhanced by the chemical bonds. Next, the results for the silicon clusters are shown in Figure 6. Again, the gain in the variance does not change with the system size and it is about 20% above the gain of 82 for a single silicon atom. Compared to carbon the gain in the variance is about 17 times larger for silicon. Note that the gain on these curves looks like to have rather large fluctuations (∼ 10%). The fluctuations come mainly from the infinite variance of the estimator of V (E L ) and we cannot rely on the central limit theorem (the rate of convergence is slower than 1/ √ M ). In Figure 5 the gain in variance is shown for the value M * s of M s which minimizes the computational cost. This gain increases from about 3

for CH 4 to about 5 for 30 carbon atoms. Silicon atom Gain V (X)/V ( X)

Silicon atoms Si n

Figure 6: The gain in the variance for alkanes of increasing length for converged sidewalks (large M s ). In dashed the gain for a converged sidewalk for a single carbon atom.

The numerical gain is much larger, it is shown in Figure 7. The speedup, i.e. the gain in computational time is the product of the gain in the variance and the gain in the correlation factor, but it includes also a reduction of the computational time of the main walk. This is because the sidewalks in the core regions take most of the information from that region, allowing to reduce the sampling of the core electrons in the main-walk (see the computational details). In practice the reduction of CPU time for a step of the main walk comes from a reduction of the acceptation probability from 0.95 to 0.57 for Alkanes and from 0.93 to 0.42 for silicons. These acceptation probabilities do not depend on the size of the molecules in the error bars which confirms that they are transferable. The corresponding gains in CPU times G 420 for large clusters of silicon. This large variance reduction for Silicon is due the large atomic charge Z but also to a choice of large cores (10 electrons). If the asymptotic limit is much better it should also be reached for much larger molecules according to Eq. ( 42), not only because r ∞ is smaller but also because t s is larger (by a factor 10). The gain we observe for the silicon cluster with 7 atoms is only 4.5, it is the same as for an alkane molecule of equal size. step. This is the standard process used in the more accurate Diffusion Monte Carlo method (DMC). [START_REF] Toulouse | Electron Correlation in Molecules -ab initio Beyond Gaussian Quantum Chemistry[END_REF] The motivation is that we want to open the extension of this work to DMC. We are introducing different time steps, two time steps τ A and τ v for the main walk and τ s for the sidewalk. The time steps τ s and τ A are small and adapted to the core electrons. The time step τ v is adapted for the valence electrons and is therefore much larger. The frequency ν A of using the small time step τ A is also a parameter to be optimized. The complete scheme is displayed in Fig. 9. A main walk with two time steps but without subsampling has not been found more efficient than a simple main walk with a single time step, this is because the fluctuations coming from the valence region are hidden by the fluctuations coming from the core. The sidewalk recovers most of the information of the core which gives us the flexibility to move the core electrons less frequently within the main walk. When resorting to the improved estimator the computational time of the main walk can be reduced because moving electrons and rejecting the moves has a O(1) CPU cost, using the algebra we develop in the appendix A, while accepting the moves implies a O(N 2 ) scaling since we update the Slater determinant with the Sherman-Morrison formula. Finally, the two time steps dy-namics enable to reduce the correlation factor but also the computational time of the main walk.

We applied the following simple protocol to obtain optimized parameters for the three time steps τ v , τ S and τ A and the ideal number of subsampling steps M s . All optimizations are carried out with the alternative small time step τ A . First, the time step of the sidewalk is determined for a sidewalk with large M s = 100 by minimizing the variance of the improved estimator and with a rough estimate of τ v and ν S = 1. Next, the time step τ v of the main walk (including subsampling) is optimized by minimizing the cost (see Eq. 10) with ν S = 1.

A single simulation with large M s allows to extract as well the results for all possible shorter sidewalks and therefore the optimization of M s . The correlation factor of the improved estimator is generally between 1-2 for these optimized parameters. Therefore, the sidewalk length M S has been determined for the subsampling frequency ν S of 1 and 2 (for the alkanes 10-16) but in all instances the former turns out to be more efficient. The parameters for molecular systems can be transferred from single atoms or small model systems e.g. the parameters for a carbon atom in arbitrary alkanes can be determined from the CH 4 molecule.

For heavier elements like silicon the time step of the subsampling is simply identical to the time step of an ideal main walk without subsampling. With the multiple time steps adapted for core and valence electrons we can reduce the correlation factor from about 4-5 for carbon or silicon with a simple main walk to about 2 with our scheme. The wavefunction Ψ comes from an SCF calculations performed with Quantum Package. [START_REF] Garniron | Quantum Package 2.0: An Open-Source Determinant-Driven Suite of Programs[END_REF] The basis is made of Slater atomic orbitals from the reference, [START_REF] Van Lenthe | Optimized Slater-type basis sets for the elements 1-118[END_REF] TZP for alkanes and SZ for silicon clusters. These atomic orbitals have been fitted using a large sum of Gaussians to be treated by Quantum Package.

Conclusion

In this work we are exploiting that most of the statistical fluctuations come from the core region and that the core regions are separable to efficiently remove the statistical fluctuations coming from the core electrons. This is done by sidewalks for each core region, i.e. moving the core electrons while freezing the environment. The computational cost is considerably reduced especially for molecular systems as each sidewalk includes only very few electrons.

The gain in computational time is about 22 for alkanes and should be 420 for large silicon clusters when computing the variational energy with a free Jastrow Slater determinant.

Overall, the presented method is a proof of concept for all-electron QMC calculations, with a numerical scaling comparable to the one obtained using pseudo-potentials. Of course the clear advantage is to avoid the related uncontrolled approximations. The obvious next step to obtain physical meaningful results is to use a Jastrow factor, adapt the scheme to other kinds of wavefunctions, and to other properties than the energy, like for example its derivatives to optimize Ψ. Note that the gain in the variance especially for large molecules with larger atoms is more significant than the gain one can obtain by improving the ergodicity of the dynamic. Very intriguing is the perspective to extend this work to Diffusion Monte Carlo (DMC).

functions are called atomic spin-orbitals. We suppose them to be localized that is they reduce to zero if the distance from a given atom is larger than a threshold. Given the configuration ω that is the N positions r i of the electrons we define X the N × p rectangular matrix of spin-orbitals.

X ij = χ j (r i ) (13)
A Slater determinant is

Φ(X) = det(XC) (14)
where C is a p × N matrix of the so-called molecular orbital coefficients. The local energy like the drift can be written as a logarithmic derivative of Φ. [START_REF] Filippi | Simple formalism for efficient derivatives and multideterminant expansions in quantum Monte Carlo[END_REF] That property holds also if the Jastrow factor is included. Here we choose to separate the kinetic energy from the local potential energy

E L = ∂ λ ln Φ X - λ 2 ∆X + ij 1 r ij + iA Z A r iA (15)
The first term is the kinetic energy, the second term is the electron-electron potential and the third is the electron-nuclei potential (Z A is the nuclear charge of the atom A). r ij is of course the distance between electron i and j while r iA is the distance between the electron i and the nucleus A. If C and X depend on a parameter λ

∂ λ ln Φ = tr(D ∂ λ X) + tr((XC) -1 X ∂ λ C) (16)
where

D ≡ C(XC) -1 (17)
represents the logarithmic gradient of Φ with respect to X. A given configuration ω defines Ω which is subsampled by moving a few electrons of ω evolving in this way to ω ∈ Ω.

Correspondingly, if X differs from X by a few lines, the determinant and its derivatives can be updated with efficient formulas. First we define the operator P which applied on the left selects those lines, P X are the lines which may differ from the lines of P X. We also define the operator Q T which applied on the right of P X or P X removes zero columns of P X and P X .

X ≡ P X Q T Using the determinant lemma Φ(X ) = det(XC) det(P X C(XC) -1 P T ) = det(XC) det(P X Q T QC(XC) -1 P T ) = det(XC) det( X C) (18)
where X and C are submatrices of X and D.

X ≡ P X Q T (19) C ≡ QC(XC) -1 P T = QDP T (20)
The second term of the r.h.s of expression (18) is a Slater determinant for the subsystem with reduced numbers of electrons and atomic orbitals, the matrix C represents effective molecular orbitals. Expr (18) performs an update of the Slater determinant for the full system using a reduced Slater determinant.

Introducing

ᾱ ≡ ( X C) -1 D ≡ C( X C) -1
the logarithmic derivative of the expression (18) is

∂ λ ln Φ(X ) = tr(D ∂ λ X) + tr( D ∂ λ X) + tr(ᾱ X ∂ λ C) = tr(D ∂ λ X) + tr( D ∂ λ X) -tr(ᾱ XQD ∂ λ X DP T )
Note that this expression does not depend on the lines of ∂ λ X which are replaced by the lines of ∂ λ X , in other words ∂ λ X can be replaced by (1 -P T P )X in Expr (21). This property can be checked algebraically and will be used later. The control variate for the local energy is

E L (X ) -E L (X) = tr D ∂ λ X -tr(ᾱ XQD ∂ λ X DP T ) + i∈Ω,j ( 1 r ij - 1 r ij ) + i∈Ω,A ( Z A r iA - Z A r iA ) (21)
where r ij and r iA represent distances from the electron i in the new configuration ω ∈ Ω.

The matrix QD ∂ λ X DP T is computed for a O(N 3 ) cost and stored once for the sidewalk.

Computing the control variate (21) has a O(N ) scaling because of the two last Coulombic terms. Indeed the index i runs only on the electrons of the subsystem Ω (core electrons), but there are N -1 electrons j and O(N ) atoms A.

We can take instead an Ω-dependent approximation of E L by considering only the interaction of the electrons in Ω with particles within a fixed distance from the center of Ω.

This reduces the Coulombic sum to a O(1) numerical cost, however we expect with little effect on the statistical fluctuations. This is because we only neglect interactions with distant particles, distant dipoles, quadrupoles or higher moments.

We propose also to remove the kinetic energy of particles which are not in the core region which defines Ω. Physically if this core region is independent of the rest of the system, we can replace the one-body terms outside Ω without modifying the difference (21). Mathematically canceling the kinetic energy outside the core region is equivalent to replacing (1 -P T P )X

C Optimal gain in efficiency/computational time

Combining Expressions (11, 12) we can write when M s is sufficiently large and the three gain factors do not depend anymore on M s

G(M s ) = G ∞ r(M s )(1 + M s ts t∞ ) (37)
Maximizing the efficiency in Eq. ( 11) is equivalent to minimizing the function

1 G(M s ) = (1 + M s t s t ∞ ) 1 + 1 M s 1 -r ∞ r ∞ cs 1 G ∞ (38)
The optimal value of M s is

M * s = t ∞ t s cs (1 -r ∞ ) r ∞ (39)
for an optimal gain G(M * s ) such that

1 G(M * s ) = 1 G ∞   1 + t s t ∞ cs (1 -r ∞ ) r ∞   2 (40)
For a large molecule ts t∞ is small and a first order Taylor expansion gives for r ∞ = 0

1 G(M * s ) = 1 G ∞   1 + 2 t s t ∞ cs (1 -r ∞ ) r ∞ + o t s r ∞ t ∞   (41)
When computing the variational energy with a Jastrow-Slater determinant and using the simplified control variate in Eq. ( 22) we have t s /t ∞ = O(N -2 ) and

1 G(M * s ) = 1 G ∞ 1 + o( 1 N √ r ∞ ) . (42)

Figure 1 :

 1 Figure 1: Schematic representation of the core subsampling.

Figure 2 : 10 Figure 3 :

 2103 Figure 2: The reduction of the variance V ( X)/V (X) as a function of the fraction of core electrons x. M s = 100 • n core for all simulations.

Figure 5 :

 5 Figure 5: The gain in the variance for alkanes of increasing length for converged sidewalks (large M s ) and for the optimized M * s . In dashed the gain for a converged sidewalk for a single carbon atom.

t

  ∞ /t are 0.95/0.57 = 1.7 for Alkanes and 0.93/0.41 = 2.3 for silicons. Finally, The correlation factors are reduced from 4.4 to 1.8 for the alkanes and 4.5 to 2.1 for the silicons which results in a gain of 2.4 and 2.1, respectively. Again, within the error bars the correlation factors do not depend on the system size and is transferable. Overall, this leads to an asymptotic value of the gain G which is 22 for carbon atoms. The numerical gain increases from G 2 for CH 4 to G 12 for 35 carbon atoms. The gain converges as O(1/N ) with the number of nuclei (see Eq. 42) and the ideal M * s increases as a linear function of N in accordance to Eq. 43. Based on the linear fit in Figure 8 for the alkane chains, M * S = 100 is for example reached for about 192 carbon atoms. One can estimate the asymptotic value of the gain

5

 5 Computational detailsA single iteration (our main walk or sidewalk) consists of a usual drift (logarithmic derivative of Ψ) and Brownian diffusion process, completed by a Metropolis acceptation-rejection

Figure 7 :

 7 Figure7: The speedup compared to a simple main walk for alkanes of increasing length. The theoretical limit (dashed line) is estimated as the product of the gain in variance and the gain in correlation factor for CH 4 .

Figure 8 :

 8 Figure 8: The ideal M * s for the alkane chains with a linear fit according to Equation 43.

Figure 9 :

 9 Figure 9: Complete scheme of the subsampling with the dual time step scheme with the main walk VMC steps (M), the subsampling steps (S) with frequency ν S and the alternative time steps (A) with frequency ν A .

Acknowledgement

J. F. acknowledges the Deutsche Forschungsgemeinschaft (DFG) for financial support (Grant FE 1898/1-1).

A Subsampling and updating Slater determinants

The wave-function is built on p functions of χ i (r) where r represents the 3 spatial coordinates of an electron and the spin (± [START_REF] Feldt | Excited-state calculations with quantum Monte Carlo[END_REF] 2 ). Because they are usually centered on each atom these by 0 in Expr (21). This leads to the (zero-expectation-value) control variate

where the sums over j and A are restricted on the electrons and the atoms in a sphere around the center of Ω. This expression is simpler as it has the same form as the expression of the local energy for the full system, and is computationally less demanding. These formulas apply with a Jastrow factor since the later only modifies the definition of the derivative

B Convergence as a function of M s , the size of a sidewalk

Given a set of random variables Ω, E(X|Ω) is an unbiased estimator of E(X) since E(X) =

E(E(X|Ω)).

Let us prove that it is a variance-reduced estimator. The conditional variance is

Now taking the expectation value of the two sides of this equation and isolating E(X) 2 on the l.h.s. we find

which becomes after removing E(X 2 ) on the two sides of this identity

The variance of the conditional estimator E(X|Ω) is then lower than V (X).

Here, X is the local energy for an atom and Ω is the set of coordinates of the valence electrons. In practice we perform a main walk and sidewalks to sample Ω i.e. moving the core electrons while freezing the valence region

For a given Ω the variance of X is

where c s is a correlation factor which takes into account that the points on a sidewalk are not independent. We assume here that c s depends only on Ω and not on M s . This property holds in a regime where M s is sufficiently large. Reminding that

We combine the two last equations and apply the expectation value

In the calculation we do not use E(X|Ω) as an improved estimator, we use instead X which converges to E(X|Ω) for large M s (ergodicity theorem). Eq. (29) tells that the variance of X converges to the variance of E(X|Ω) hyperbolically. The variance of X is a fraction

which becomes the full gain only in the limit M s → ∞. Introducing the mean correlation

Equation (29) becomes

where we used Eq. ( 24) for the last expression. A hyperbolic fit of the function r(M s ) can provide the two parameters r ∞ and cs . One can also use two values M s and αM s

The explicit dependence on M s should not make forget that cs is converging to a constant when M s is sufficiently large.

Regarding the corresponding optimal value of M s the Taylor expansion gives

which means that the number of side-steps (of the subsampling) for each core is optimally proportional to the number of cores.