Prof. Dr Cornelia M Van Duijn
email: c.vanduijn@erasmusmc.nl

Ayse Demirkan

Najaf Amin

Aaron Isaacs

Marjo-Riitta Jarvelin

John B Whitfield

Heinz-Erich Erich Wichmann

Kirsten Ohm Kyvik

Igor Rudan

Christian Gieger

Ayşe Demirkan

Andrew A Hicks

Åsa Johansson

Jouke-Jan Hottenga

Johannes J Smith

Sarah H Wild

Nancy L Pederson

G Willemsen

Massimo Mangino

Caroline Hayward

André G Uitterlinden

Albert Hofman

Jacqueline Witteman

Grant W Montgomery

Kirsi H Pietiläinen

Taina Rantanen

Jaakko Kaprio

Angela Döring

Peter P Pramstaller

Ulf Gyllensten

Eco J C De Geus

Brenda W Penninx

James F Wilson

Fernando Rivadeneria

Patrik K E Magnusson

Dorret Boomsma

Tim Spector

Harry Campbell

Birgit Hoehne

Nicholas G Martin

Ben A Oostra

Mark Mccarthy

Leena Peltonen- Palotie

Yurii Aulchenko

Peter Visscher

Samuli Ripatti

A Cecile

J W Janssens

Genetic architecture of circulating lipid levels

Keywords: Serum lipids, polygenic, genome-wide association, polygenic score, pathway analysis

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Introduction

Serum concentrations of low-density lipoprotein cholesterol (LDL-C), high-density lipoprotein cholesterol (HDL-C), triglycerides (TG) and total cholesterol (TC) are highly heritable phenotypes associated with the risk of cardiovascular morbidity and mortality [START_REF] Isaacs | Heritabilities, apolipoprotein E, and effects of inbreeding on plasma lipids in a genetically isolated population: the Erasmus Rucphen Family Study[END_REF][START_REF] Kannel | 3rd: Factors of risk in the development of coronary heart disease--six year follow-up experience. The Framingham Study[END_REF][START_REF] Kuulasmaa | Estimation of contribution of changes in classic risk factors to trends in coronary-event rates across the WHO MONICA Project populations[END_REF][START_REF] Namboodiri | The Collaborative Lipid Research Clinics Program Family Study. IV. Familial associations of plasma lipids and lipoproteins[END_REF] . A number of genome-wide association studies (GWAS) successfully identified multiple genes influencing circulating lipid levels [START_REF] Aulchenko | Loci influencing lipid levels and coronary heart disease risk in 16 European population cohorts[END_REF][START_REF] Kathiresan | Six new loci associated with blood lowdensity lipoprotein cholesterol, high-density lipoprotein cholesterol or triglycerides in humans[END_REF][START_REF] Kathiresan | Common variants at 30 loci contribute to polygenic dyslipidemia[END_REF][START_REF] Kooner | Genome-wide scan identifies variation in MLXIPL associated with plasma triglycerides[END_REF][START_REF] Sabatti | Genome-wide association analysis of metabolic traits in a birth cohort from a founder population[END_REF][START_REF] Sandhu | LDL-cholesterol concentrations: a genome-wide association study[END_REF][START_REF] Wallace | Genome-wide association study identifies genes for biomarkers of cardiovascular disease: serum urate and dyslipidemia[END_REF][START_REF] Willer | Newly identified loci that influence lipid concentrations and risk of coronary artery disease[END_REF] . There are currently over 100 established loci that include both common variants with relatively small effects as well as a considerable number of rare variants with large effects [START_REF] Hegele | Plasma lipoproteins: genetic influences and clinical implications[END_REF] . Despite these successes, a substantial proportion of the heritability of each trait remains unexplained, suggesting that many determinants have yet to be identified [START_REF] Manolio | Finding the missing heritability of complex diseases[END_REF] .

Several plausible explanations may underlie the unexplained heritability of lipid traits, including the presence of both unknown common variants with small effects and novel rare variants with larger effects. The ENGAGE GWAS [START_REF] Aulchenko | Loci influencing lipid levels and coronary heart disease risk in 16 European population cohorts[END_REF] was one of the first large population based studies designed to find variants associated with circulating lipid levels. The study, based on 16 European cohorts including up to 22562 individuals, identified 6 novel loci, in addition to replicating 16 previously known loci. However, as demonstrated by the recent GWAS from the Global Lipid Genetics Consortium (GLGC), numerous additional variants passed the genome-wide significance threshold as a result of increased sample size [START_REF] Teslovich | Biological, clinical and population relevance of 95 loci for blood lipids[END_REF] . The GLGC GWAS, which included over 100000 individuals of European ancestry, reported 95 loci, with 59 reaching genome-wide significance for the first time. These results raise an interesting question: if common variants remain to be discovered, how many should we expect? Are there still a limited number of loci or can we expect a polygenic mechanism that involves a very large number of variants with very small effects? In the latter case, these variants would contribute to a continuous spectrum of alleles spanning the genome and single genes involved in this complex polygenic model might not be detectable by GWAS, regardless of sample size [START_REF] Visscher | Sizing up human height variation[END_REF] . Evidence for this type of genetic architecture can be shown using a genome-wide scoring approach, as was recently demonstrated for a number of psychiatric outcomes [START_REF] Demirkan | Genetic risk profiles for depression and anxiety in adult and elderly cohorts[END_REF][START_REF] Purcell | Common polygenic variation contributes to risk of schizophrenia and bipolar disorder[END_REF][START_REF] Amin | Genetic scoring analysis: a way forward in genome wide association studies?[END_REF] . Additionally, these polygenic scores may provide extra information useful in determining p-value thresholds for pathway analysis.

The current study aimed to explore the extent to which common variation accounts for the unexplained heritability of circulating lipid levels using the genome-wide scoring method.

We also evaluated the evidence for a common polygenic effect underlying different lipid traits, using the same risk scoring approach. Finally, we examined the utility of genome-wide polygenic scores for identifying pathways beyond those identified using a classical GWAS approach.

Materials and methods

The polygenic risk score approach involves using results from a discovery set to explore the genetic architecture of an independent target sample. Our discovery set consisted of the metaanalysis of 16 European populations from the ENGAGE Lipid Consortium (N = 17798 -22562) (Table 1). A detailed description of this study, including populations, genotyping information and statistical analysis, was previously published [START_REF] Aulchenko | Loci influencing lipid levels and coronary heart disease risk in 16 European population cohorts[END_REF] .

The target sample consisted of RS-II, an extension of the Rotterdam Study, a prospective cohort study started in 1990 in the Ommoord district of the city of Rotterdam. RS-II, which was not a part of the ENGAGE discovery set, consists of 3011 participants (out of 4472 invitees) who were 55 years or older during the recruitment period (2000-2001) [START_REF] Hofman | The Rotterdam Study: 2010 objectives and design update[END_REF] . Of the 3011, 2540 persons were successfully genotyped with an Illumina 610K array. Fasting HDL-C, TG and TC were measured with enzymatic colorimetric tests on a Roche/Hitachi 911 analyzer (Roche Diagnostics, Meylan, France). LDL-C was estimated using the Friedewald formula [START_REF] Friedewald | Estimation of the concentration of lowdensity lipoprotein cholesterol in plasma, without use of the preparative ultracentrifuge[END_REF] .

SNPs included in the construction of the polygenic scores were based on the results from the ENGAGE study. We selected different clusters of SNPs for the calculation of the scores using several p-value thresholds (p discovery ) ranging from 5 ×10 -8 to 0.5. We calculated genetic scores for those various clusters of SNPs in the target sample by multiplying the number of risk alleles for each SNP (0, 1 or 2) by the effect sizes from the discovery set, and summing them up across all the SNPs in that cluster. For this analysis we used the PLINK "profile scoring" option. SNPs that had a call rate < 90% or HWE p-value < 1×10 -8 were excluded from these computations. A/T and G/C polymorphisms were also excluded to avoid potential strand inconsistencies. SNPs in linkage disequilibrium (LD) were pruned over 200 SNP sliding windows using a pair wise r 2 threshold of 0.25 in PLINK [START_REF] Purcell | PLINK: a tool set for whole-genome association and population-based linkage analyses[END_REF] . LD pruning was performed per SNP cluster. (See Supplementary Table 2 for the number of SNPs remaining in each cluster and used for analysis)

The associations between these scores and serum lipid levels were tested in SPSS using linear regression models with sex, age and age 2 as covariates (the same covariates as included in the discovery GWAS). The proportion of total variance explained by the genetic score, referred to here as the percentage of explained variance (PEV), was determined by comparing models with/without the risk score.

To evaluate whether the PEV results were driven by the GWAS hits, we also constructed a variable comprised of only the significant GWAS variants and included it as a covariate in our original models. When calculating the polygenic scores for these analyses, we also removed SNPs within 2 Mb windows surrounding the GWAS hits. We employed exactly the same pruning approach for this analysis.

In order to search for evidence for a shared genetic background between various lipid traits, we tested additional models in which we used the polygenic score for a particular lipid and tried to predict the others, for instance, utilizing the HDL-C polygenic score to predict TG, and vice versa.

The score which yielded the highest PEV for a given lipid trait ostensibly includes the most valuable genetic information; therefore we selected these thresholds to utilize in pathway analysis (in contrast to using only genome-wide significant loci). For these analyses, we used the PANTHER tools (http://www.pantherdb.org) [START_REF] Thomas | PANTHER: a browsable database of gene products organized by biological function, using curated protein family and subfamily classification[END_REF] . We first tested the genome-wide significant SNPs (p discovery < 5 × 10 -8 ) from the ENGAGE GWAS in the pathway analysis.

These results were then compared with those obtained using alternative p discovery thresholds selected on the predictive ability of the polygenic scores. After SNP selection, SNPs within gene regions were converted to gene symbols using the "SCAN SNP and CNV annotation database" (http://www.scandb.org). Gene lists were tested for enrichment in three PANTHER categories: (1) pathways, (2) biological processes and (3) molecular functions. Testing for enrichment basically involves comparing one gene list to the reference list to statistically determine over-or under-representation of PANTHER classification categories. Based on the reference list, an expected value is computed (the number of genes one would expect in the list for a particular PANTHER category) and it is assumed that, under the null hypothesis, genes in the tested list are sampled from the same distribution as genes from the reference set.

The Homo sapiens gene list from National Center for Biotechnology Information was used as the reference gene list. To avoid bias caused by multiple testing, PANTHER's Bonferroni correction option was implemented. (See Supplementary Figure 1 for the over all flowchart of the study)

Results

Table 1 shows summary statistics for the discovery and target samples. The female/male ratio in the discovery set was significantly higher compared to the target set (1.6 vs 1.2, p < 0.001). Genome-wide significant SNPs from the ENGAGE GWAS were checked for their associations in the target sample using linear regression. Generally, evidence of association between those SNPs and lipid levels were marginally significant or nonsignificant (Supplementary Table 1). The GWAS of circulating lipids in RS-II did not show any genome-wide significant findings except the CETP gene region SNPs which were associated to HDL-C (rs7499892, p=3.4 ×10 -13 ). Manhattan plots for the GWAS of the HLD-C, LDL-C, TG and TC can be found in the Supplementary Figure 2.

Prediction

Figure 1 shows the PEV obtained for each lipid trait using the polygenic scores generated for a number of p-value thresholds in the target sample (RS-II). For HDL-C, the polygenic score computed using 19 genome-wide significant SNPs from 8 gene regions (p discovery < 5 ×10 -8 ) resulted in the maximum PEV compared to the null model (4.75 %, p = 3.6 ×10 -30 ; Figure 1 A). For LDL-C, (Figure 1 B), the maximum PEV was observed with the polygenic score that included 21 SNPs with a p discovery < 1×10 -6 (2.6 %, p = 5.1 × 10 -16 ). Figure 1 C shows PEVs for TG levels; the score that included 12 SNPs from 8 regions with p discovery < 1×10 -7 (3.8 %, p = 2.8 × 10 -21 ) was the best predictor. For these traits, the variance explained decreased with the inclusion of additional SNPs in the polygenic score selected using more liberal p discovery thresholds (Figure 1 A to C). Finally, for TC, the highest PEV was obtained using 46 SNPs from 24 regions with p discovery < 10 -5 (2.7%, p = 1.4 × 10 -16 ). This was higher than the PEV obtained using only the genome-wide significant SNPs (PEV = 2.1%, p = 8.2 × 10 -13 , n=20 SNPs from 11 regions; Figure 1 D). As with HDL-C, LDL-C, and TG, the explained variance for TC dropped when more liberal p discovery thresholds were used to construct the polygenic score. For LDL-C, HDL-C and TC, all scores were significant (up to a threshold of p discovery < 0.5). We observed similar patterns when we used unpruned data (Supplementary Figure 3).

Figure 2 shows the results from the second approach, in which models were adjusted for genome-wide significant variants. For HDL-C (Figure 2 A), the PEV increased as SNPs were added, up to 0.5 % with p discovery < 0.1 (p= 1.0 × 10 -4 ) and remained significant until p discovery < 0.5 (p=2.3 × 10 -4 ). A similar pattern was observed with LDL-C (Figure 2 B, explained variance was up to 0.4 % (p= 0.002) with p discovery threshold of 0.2. In contrast, the polygenic score for TG, when the effects of known variants were excluded, was not associated with TG levels in the target population (Figure 2 C). For TC (Figure 2 D), the maximum PEV was observed with p discovery < 1×10 -5 , (0.6 % and p= 1.8 × 10 -4 ).

Cross prediction

Table 2 shows the phenotypic correlations for the four outcomes studied, and additionally shows the correlations between the polygenic scores for different p discovery thresholds.

Correlations between the traits were modest, with the exceptions of TC and LDL-C, (r = 0.9) and TG and HDL-C (r = -0.5). The correlations between the polygenic scores were weaker than the phenotypic correlations (0.8 for TC/LDL-C and -0.2 for TG/HDL for p discovery < 5 × 10 -8 )

To evaluate the evidence for common polygenetic effects underlying lipid levels, we performed cross-prediction analyses (Figure 3). The highest PEV was based on the TC score at p discovery (TC) < 1×10 -5 , which explained up to 2.7% of the variance in circulating LDL-C (p = 2.0 × 10 -5 ; Figure 3 K). Similarly, LDL-C risk profiles explained up to 1.8 % of the variance in TC when we selected all SNPs with a p discovery (LDL-C) < 10 -6 (p = 1.4 × 10 -11 ; Figure 3 F).

These findings are in line with the high phenotypic correlations between those variables.

Pathway analysis

Pathways analyses using only genome-wide significant SNPs was compared to the analogous analyses using SNPs from the polygenic scores which yielded the highest PEV for each trait (Figure 1). These scores used thresholds of p < 1 ×10 -6 for LDL-C, p < 1 ×10 -5 for TC, p < 5 × 10 -8 for HDL-C and p < 1 ×10 -7 for TG. Table 3 shows the findings from the pathway analysis, based on alternatives to a p-value threshold of 5.0 × 10 -8 . None of the pathways among categories defined by the PANTHER tool were significant after strict adjustment for multiple testing (Bonferroni correction). With respect to biological processes the lipid and fatty acid transport, and lipid, fatty acid and steroid metabolism pathways were two biological processes enriched in the HDL-C and LDL-C GWAS findings. At the level of molecular function, genes with an apolipoprotein and transfer/carrier function were enriched in LDL-C, while genes with a lipase function were observed to be significantly enriched among the top GWAS results for HDL-C. For HDL-C and TG, we were not able to select alternative p-value thresholds since the highest PEVs were observed with p < 5 × 10 -8 . With respect to LDL-C, the pathway analysis utilizing two different p-value thresholds (p < 1×10 -6 and p < 5 × 10 -8 ) resulted in the same findings. No additional pathways were identified by using extra information from the risk profiles for LDL-C, TG and HDL-C. For TC, on the other hand, the lipid, fatty acid and steroid metabolism, lipid and fatty acid transport and transport terms additionally emerged among biological processes tested using the alternative threshold (Table 3).

Discussion

Using prediction modelling, we could explain up to 4.8% of the variance in HDL-C, 2.6% in LDL-C, 3.8% in TG and 2.7% in TC. These PEVs are very similar to those from similar studies [START_REF] Aulchenko | Loci influencing lipid levels and coronary heart disease risk in 16 European population cohorts[END_REF][START_REF] Sabatti | Genome-wide association analysis of metabolic traits in a birth cohort from a founder population[END_REF] and much higher than the single SNP analysis of genome-wide significant SNPs from the ENGAGE GWAS (Supplementary Table 1) However, these proportions are much lower than those identified by GLGC, which were estimated to explain 12.4% (TC), 12.2% (LDL-C), 12.1% (HDL-C), and 9.6% (TG) of the variance in the Framingham Heart Study sample, as mentioned by Teslovich et al. [START_REF] Teslovich | Meta-analysis of >100,000 individuals identifies 63 new loci associated with serum lipid concentrations: The American Society of Human Genetics[END_REF] . This is expected since increases in sample size lead to better estimation of the effect sizes of the SNPs and GLGC had a sample size 5 times larger than the ENGAGE sample, which we used as a discovery set in our study.

For all of the traits, the PEV reached a maximum and then decreased with the use of more liberal p discovery thresholds to calculate the polygenic scores (Figure 1). This is most likely explained by the inclusion of more and more biologically non-relevant SNPs, so that the effects of true positive findings are diluted and this is reflected by the decreases in PEV.

For all of the studied traits, we found the highest PEV when the polygenic score was based on SNPs with a low p discovery value (5 × 10 -8 for HDL-C, 1×10 -7 for TG, 1×10 -6 for LDL-C and 1×10 -5 for TC).

Including the top regions from the ENGAGE GWAS dataset as a separate predictor in the models (Figure 2) uncovered a residual polygenic component which does not explain more than 1% of HDL-C, LDL-C and TC levels. These findings suggest that there are unknown genes with much smaller effects involved in determining these outcomes. However, the PEVs for these additional variants were small when compared with those for the top findings. For TG, on the contrary, excluding the top regions from the polygenic score resulted in non-significant findings. For TC, which is highly heterogeneous compared to the other traits, it seems that some variants remain to be discovered ( p discovery < 1×10 -5 ).

It is of note that among newly discovered loci for HDL-C by GLGC, leading SNPs from 10 loci had p values >0.05 in the ENGAGE HDL-C analysis. Similar findings were observed for 10 loci for LDL-C, 3 loci for TG and for 9 loci in TC [START_REF] Teslovich | Meta-analysis of >100,000 individuals identifies 63 new loci associated with serum lipid concentrations: The American Society of Human Genetics[END_REF] .

It is already known that monogenic disorders [START_REF] Rader | Monogenic hypercholesterolemia: new insights in pathogenesis and treatment[END_REF] and rare variants also account for variation in circulating lipid levels [START_REF] Cohen | Multiple rare alleles contribute to low plasma levels of HDL cholesterol[END_REF][START_REF] Cohen | Multiple rare variants in NPC1L1 associated with reduced sterol absorption and plasma low-density lipoprotein levels[END_REF][START_REF] Gilbert | Lipoprotein lipase (LPL) deficiency: a new patient homozygote for the preponderant mutation Gly188Glu in the human LPL gene and review of reported mutations: 75 % are clustered in exons 5 and 6[END_REF][START_REF] Kotowski | A spectrum of PCSK9 alleles contributes to plasma levels of low-density lipoprotein cholesterol[END_REF][START_REF] Simha | Inherited lipodystrophies and hypertriglyceridemia[END_REF][START_REF] Slatter | Novel rare mutations and promoter haplotypes in ABCA1 contribute to low-HDL-C levels[END_REF][START_REF] Talmud | Rare APOA5 mutations--clinical consequences, metabolic and functional effects: an ENID review[END_REF] . This may help to explain why the explained variance is small compared to the high heritability of the traits, especially since many rarer variants are population specific, and might not have been well-represented in our European dataset, or not well tagged by the common SNPs under study. For instance, APOE gene variations are tagged by the CEACAM16-TOMM40 region among the ENGAGE GWAS top findings, and SNPs from this region were not associated to LDL-C levels in RS-II, however, APOE ε2 carrier status explains 2.6 % of the phenotypic variation in LDL-C levels in RS-II.

Additionally, the gender ratio difference between the discovery and target samples may have been a limitation to the current study, since some loci show different effect sizes for males and females [START_REF] Aulchenko | Loci influencing lipid levels and coronary heart disease risk in 16 European population cohorts[END_REF] .

Our findings have implications for gene discovery and suggest that GWAS of much larger samples may be needed to discover additional variants with small effects for HDL-C and LDL-C. However, at the same time, this study suggests that many of the unknown SNPs have relatively large effects and that is confirmed by the GLGC data. Our findings suggest that GWAS on serum lipids in the future will still be successful as sample sizes increase [START_REF] Manolio | Finding the missing heritability of complex diseases[END_REF] .

Our cross prediction results are interesting from a biological perspective. These findings showed very little overlap between the polygenic scores for different circulating lipids. A strong inverse relationship exists between low HDL-C and elevated plasma TG (r = -0.5 in RS-II). Low HDL-C levels are strongly associated with hypertriglyceridemia since high levels of plasma triglycerides drive an exchange reaction for HDL-C cholesteryl esters mediated by CETP [START_REF] Genest | Familial lipoprotein disorders in patients with premature coronary artery disease[END_REF] . In addition, the triglyceride and phospholipids in HDL-C are hydrolyzed by LIPC [START_REF] Hegele | Plasma lipoproteins: genetic influences and clinical implications[END_REF][START_REF] Genest | Familial lipoprotein disorders in patients with premature coronary artery disease[END_REF] . However, using our genetic evaluation it was not possible to predict a large proportion of the variance in TG levels using HDL-C risk profiles despite the correlation between the two lipids. The polygenic score for TG was slightly better in predicting HDL-C than when we used the top SNPs, however, the PEV did not exceed 0.6 % and was lower than the variance explained by HDL-C SNPs and also lower than the variance explained in circulating TG by TG SNPs. Thus, our data implies that common genetic variants involved in determining both TG and HDL-C levels do not explain the phenotypic correlation between these traits, suggesting that the correlation may be influenced strongly by environmental factors, and/or restricted to a few genes. An alternative explanation may be that we tested the polygenic effects of common variants weighted by their effect size from the initial GWAS. When there are strong causal variants among the top hits that are specific to HDL-C but not to TG, this may dilute the effect of genes with small effect sizes on both outcomes. Also, the current analyses do not account other forms of genetic variation, such as rare variants or copy number variations (CNVs).

As expected, we also found evidence for a number of genes that regulate both HDL-C and LDL-C (Figure 3A) and a similar overlap between TG and LDL-C (Figure 3 H). TC SNPs were able to explain up to 2.7% of the variation in LDL-C, suggesting that the genes determining LDL-C and TC are for a large part overlapping. This result is in-line with the high phenotypic correlation between the two measures. Genome-wide significant findings from the ENGAGE GWAS harboured two loci (apolipoprotein B and LPL) influencing both HDL-C and TG, 2 loci influencing both TG and TC (DOCK7 and CEACAM16-TOMM40 regions) and 7 loci influencing both LDL-C and TC (CELSR2, APOB, ABCG5, HMGCR, FADS2/3, LDLR and CEACAM16-TOMM40). A limitation here is that LDL-C was not directly measured but calculated with the Friedewald formula in the RS-II sample and so, by definition, depends directly on TC, HDL-C and TG. This may cause a potential bias in findings for LDL-C and may inflate the association between lipids in cross prediction findings with this phenotype. We investigated whether the polygenic score approach can be used as a tool for selecting SNPs of interest in order to further evaluate them in a pathway analysis.

First, we evaluated the genome-wide significant SNPs from an existing GWAS and compared the results to those obtained using the SNPs from the polygenic model with the maximum PEV. Neither of the approaches yielded any novel pathways/biological processes (only those already known to be involved in lipid metabolism, such as cholesterol biosynthesis; lipid and fatty acid transport; and lipid, fatty acid and steroid metabolism). Also, we see that, although the use of the polygenic score approach did not provide extra information concerning LDL-C, HDL-C or TG, for TC, pathway analysis based on the best predicting polygenic score (with p discovery <1×10 -5 ) was more informative than analysis based solely on the genome-wide significant findings. Including TC SNPs up to a more liberal threshold of 1×10 -5 suggested three processes which are already biologically known but were not detectable with the 5 × 10 - positives that are not significant at the genome-wide level. Taken together with the polygenic component analysis results, it is likely that ENGAGE TC-GWAS results harbor undiscovered associated variants distributed between 1×10 -6 < p discovery < 1×10 -5 .

Using a gene scoring approach, we tested the evidence of a polygenic component for the heritable circulating lipids. We concluded that a polygenic form of inheritance exists for HDL-C, LDL-C, TG and TC. These findings may be useful for future gene discovery efforts for lipids. We also tested for possible genetic overlap between biologically related lipid traits and compared two different approaches for pathway analysis. This study gives an example of utilizing the risk scoring approach to search for the common genetic background of different quantitative traits, thus; it may also be an example for more sophisticated future studies. Biology (CMSB). We are grateful to all study participants and their relatives, general practitioners and neurologists for their contributions and to P. Veraart for her help in genealogy, J. Vergeer for the supervision of the laboratory work and P. Snijders for his help in data collection. 1 Table 3 Pathway analysis Enrichment of a particular "pathway", "biological process" or "molecular function" PANTHER categories were tested by pathway analysis. SNPs that are included in the pathway analysis are selected based on their p discovery values which were 10 -6 for LDL-C, 10 -5 total cholesterol, 5 × 10 -8 for HDL-C and 10

Figures 3 G

 3 Figures 3 G to I show the predictions based on a TG score which explained up to 0.8 % of the

Table 1 Descriptive data of discovery and replication samples

 1

		ENGAGE	RS-II
		Men	Women	Men	Women
	Number of subjects	8403	14159	1061	1253
	HDL-C (mmol/L)	1.3 (0.3)	1.6 (0.4)	1.2 (0.3)	1.5 (0.4)
	LDL-C (mmol/L)	3.4 (0.9)	2.3 (0.9)	3.6 (0.9)	1.5 (0.8)
	TG (mmol/L)	1.6 (1.1)	1.1 (0.7)	1.6 (0.9)	1.5 (0.8)
	TC (mmol/L)	5.6 (0.9)	5.9 (0.9)	5.6 (1.1)	5.7 (1.1)

Table 2 Correlation matrix of circulating lipids and genetic risk scores in RS-II.

 2 Lower-left side of the matrix shows the phenotypic correlation between circulating lipid levels, adjusted by age, age 2 and sex. Upper-right side of the matrix shows the correlation between the genetic risk scores of four circulating lipids, for the first four risk scores with p discovery < 5 × 10 -8 , p discovery < 1 × 10 -7 , p discovery < 1 × 10 -6 and p discovery < 1 × 10 -5 . * Correlation significant at p<0.05. **Correlation significant at p<0.001.

			5 ×10 -8	HDL-C	LDL-C 0.01	TG -0.20 **	TC 0.02
	Correlation between the phenotypes	HDL-C LDL-C TG	1 ×10 -7 1 ×10 -6 1 ×10 -5 5 ×10 -8 1 ×10 -7 1 ×10 -6 1 ×10 -5 5 ×10 -8 1 ×10 -7 1 ×10 -6 1 ×10	-0.1 ** -0.5 **	0.01 -0.01 0.02 0.1 **	-0.17 ** -0.09 ** -0.04 * 0.01 0.02 0.05 * 0.03	0.03 0.07 ** 0.05 * 0.76 ** 0.75 ** 0.81 ** 0.71 ** 0.13 ** 0.13 ** 0.12 **	Correlation between the genetic risk scores

  -7 for Triglycerides. NCBI: Number of genes that belong to the particular category. Observed:Number of genes that belong to the given particular category among GWAS results. Expected:Expected value for number of genes that belong to the particular pathway among GWAS results. Over/Under: Stands for "overrepresented /underrepresented". * p-value corrected for multiple testing. n.s.: no significant findings.

			NCBI	Observed Expected Over/Under	p	p*
	PATHWAYS n.s
	BIOLOGICAL PROCESS
	HDL-C	Lipid, fatty acid and steroid metabolism Lipid and fatty acid transport	770 131	5 3	0.42 0.07	+ +	4.05 ×10 -5 1.26 ×10 -3 4.77 ×10 -5 6.91 ×10 -3
	LDL-C	Lipid, fatty acid and steroid metabolism Lipid and fatty acid transport	770 131	4 3	0.51 0.09	+ +	1.46 ×10 -3 4.52 ×10 -2 8.81 ×10 -5 1.28 ×10 -2
	TG	n.s
	TC	Lipid, fatty acid and steroid metabolism Lipid and fatty acid transport Transport	770 131 1306	6 4 8	1.21 0.21 2.05	+ + +	1.22 ×10 -3 3.78 ×10 -2 5.55 ×10 -5 8.05 ×10 -3 8.47 ×10 -4 2.63 ×10 -2
	MOLECULAR FUNCTION HDL-C Lipase LDL-C Apolipoprotein Transfer/carrier protein	75 23 327	3 2 3	0.04 0.02 0.22	+ + +	9.11×10 -6 1.47 ×10 -3 1.10 ×10 -4 1.77 ×10 -2 1.26 ×10 -3 3.66 ×10 -2
	TG	n.s
	TC	n.s

discovery threshold. This finding shows that for complex traits like TC, the risk scoring approach might be used to select the SNP cluster which harbours a large number of true

Acknowledgements

ORCADES was supported by the Chief Scientist Office of the Scottish Government, the Royal Society and the European Union framework program 6 EUROSPAN project (contract no. LSHG-CT-2006-018947). DNA extractions were performed at the Wellcome Trust Clinical Research Facility in Edinburgh. We would like to acknowledge the invaluable contributions of Lorraine Anderson and the research nurses in Orkney, the administrative team in Edinburgh and the people of Orkney. For the MICROS study, we thank the primary care practitioners Raffaela Stocker, Stefan Waldner, Toni Pizzecco, Josef Plangger, Ugo Marcadent and the personnel of the Hospital of Silandro (Department of Laboratory Medicine) for their participation and collaboration in the research project. In South Tyrol, the study was supported by the Ministry of Health and Department of Educational Assistance, University and Research of the Autonomous Province of Bolzano and the South Tyrolean Sparkasse Foundation. Genome-wide genotyping of the Rotterdam Study was supported by NWO (175.010.2005.011). The ERF study was supported by grants from The Netherlands Organisation for Scientific Research, Erasmus MC and the Centre for Medical Systems


Christoph Albermann
email: christoph.albermann@imb.uni-stuttgart.de

High versus low level expression of the lycopene biosynthesis genes from Pantoea ananatis in Escherichia coli

Keywords: chromosomal integration, heterologous carotenoid production, lycopene, tetradehydrolycopene

The heterologous synthesis of lycopene in non-carotengenic Escherichia coli required the introduction of the biosynthesis genes crtE, crtB, and crtI. Recombinant E. coli strains, expressing each lycopene biosynthesis gene from Pantoea ananatis using multi-copy plasmid or single-copies after stable chromosomal integration, were cultivated and the formation of lycopene was investigated. The different expression conditions significantly influenced the lycopene formation as well as the growth behaviour. High plasmid expression levels of crtI with a single copy background of crtE and crtB in E. coli led to a predominate synthesis of tetradehydrolycopene at 253 µg g -1 (cdw).

Introduction

Carotenoids are ubiquitous with important nutritional and physiological relevance for plants, algae and bacteria. Carotenoids have gained industrial importance as colorant for cosmetic-, food-, and animal feed-products. Recent reports on carotenoids have further demonstrated that carotenoids have manifold beneficial effects on human health [START_REF] Nagao | Absorption and function of dietary carotenoids[END_REF]. The simplest C 40 carotenoid, lycopene, is an effective antioxidant and has promising biological activity for the prevention of several types of cancer, including breast cancer and prostate cancer [START_REF] Seren | Lycopene in cancer prevention and treatment[END_REF]. The genes responsible for the synthesis of lycopene have been isolated from the epiphytic bacterium, Erwinia, and plants, and the functions of the carotenogenic genes have been identified [START_REF] Hundle | Carotenoids of Erwinia herbicola and an Escherichia coli HB101 strain carrying the Erwinia herbicola carotenoid gene cluster[END_REF][START_REF] Misawa | Elucidation of the Erwinia uredovora carotenoid biosynthetic pathway by functional analysis of gene products expressed in Escherichia coli[END_REF][START_REF] To | Analysis of the gene cluster encoding carotenoid biosynthesis in Erwinia herbicola Eho13[END_REF][START_REF] Albrecht | Molecular cloning and functional expression in E. coli of a novel plant enzyme mediating zeta-carotene desaturation[END_REF][START_REF] Sandmann | Carotenoid biosynthesis in microorganisms and plants[END_REF]). The first substrate of the enzymes encoded by the Erwinia carotenoid biosynthesis gene cluster is farnesyl pyrophosphate (FPP) which is the common precursor for the biosynthesis of carotenoids and also of numerous other isoprenoids. The biosynthesis of lycopene in E. coli requires the expression of the genes crtE, crtB, and crtI, which utilize the E. coli isoprenoid precursor FPP. CrtE is a geranylgeranyl synthase that catalyzed the condensation of FPP and IPP to GGPP and CrtB is a phytoene synthase [START_REF] Sandmann | New functional assignment of the carotenogenic genes crtB and crtE with constructs of these genes from Erwinia species[END_REF]). The phytoene desaturase (CrtI) catalyzed the introduction of double bonds leading to an extended conjugated double bond system. The insertion of four double bonds into phytoene by CrtI from Pantoea ananatis (formerly Erwinia uredovora) results in the formation of lycopene [START_REF] Sandmann | Evolution of carotene desaturation: the complication of a simple pathway[END_REF].

Due to the increasing demand for carotenoids, improvements have been made in the formation of carotenoids by recombinant bacteria, yeasts, and plants expressing carotenoid biosynthesis genes or gene clusters [START_REF] Das | An update on microbial carotenoid production: application of recent metabolic engineering tools[END_REF]. In particular, the non-carotenogenic E. coli has been widely used as a host for improved carotenoid productivity by transformation with appropriate biosynthesis genes, and methods were developed for increasing the biosynthesis of isoprenoid precursors in E. coli [START_REF] Harada | Novel approaches and achievements in biosynthesis of functional isoprenoids in Escherichia coli[END_REF].

In this study, the formation of lycopene by heterologous E. coli strains expressing each biosynthesis gene from P. ananatis either from a multi-copy plasmid or from single copy inserted into the chromosome is presented.

Material and methods

Bacterial strains, growth conditions, and media

The bacterial strains used were Escherichia coli BW25113 lac + and E. coli DH5Table 1.

If not stated otherwise, they were grown at 37 °C in LB medium. Antibiotics used were: 100 μg ampicillin/ml, 50 μg chloramphenicol/ml and 50 µg kanamycin/ml. Difco MacConkey agar was purchased from Nordwald (Hamburg, Germany). Carotenoid standards were provided by the Institute of Technical Biochemistry, University of Stuttgart. All other chemical and reagents are from Sigma-Aldrich or Roth.

Cloning of the biosynthesis genes in E. coli

In order to create an artificial lycopene gene cluster for expression in E. coli, the following cloning strategy was used. Modified PCR primers were used to create a BglII restriction site upstream the start-codon of the gene, plus an artificial RBS and a BamHI or a BglII site downstream of the stop-codon. crtB was amplified from plasmid pCAR16 [START_REF] Misawa | Elucidation of the Erwinia uredovora carotenoid biosynthetic pathway by functional analysis of gene products expressed in Escherichia coli[END_REF] verify the orientations and sequences of the cloned biosynthesis genes. Takara PrimeStar DNA polymerase (Lonza, Belgium) was used for amplification. The chromosomal integration of recombinant crt-gene expression cassettes and elimination of antibiotic resistants gene cassettes using plasmid pCP20 [START_REF] Cherepanov | Gene disruption in Escherichia coli: Tc R and Km R cassettes with the option of Flp-catalyzed excision of the antibiotic-resistance determinant[END_REF] were conduct as described by [START_REF] Albermann | A simple and reliable method to conduct and monitor expression cassettes integration into the Escherichia coli chromosome[END_REF].

Cultivation of E. coli cultures and analysis of carotenoids

Recombinant, lycopene-producing E. coli strains were cultivated at 30°C in 50 ml LB medium. Plasmid carrying strains were cultivated in the presents of a corresponding antibiotic. IPTG was added at an OD 600 of 0.3. Cultures with strains that carry plasmid pJOE5559 (unpublished work) were also induced with L-rhamnose at 1.2 mM. Samples (2 ml) were withdrawn from cultures after 24h. The cells were harvested by centrifugation, washed with cold water, and subsequently extracted with acetone (500 µl) for 15 min at 50°C.

Insoluble components of the extract were removed by centrifugation (20,000 x g). HPLC

analysis were performed on Dionex HPLC Instrument installed with a Chromeleon Software, Gina autosampler, P580 pumps, and detection with a UV lamp. Products were analyzed by loading 20 µl supernatant onto a C30-reverse-phase HPLC column (250 mm x 4.6 mm, 5 µm, YMC-Europa GmbH, Dinslaken, Germany) attached with a guard column containing matrix of same material as the column. The solvents used (at 1 ml/min) were A: tert-butyl ether (MTBE)/methanol/water (19/80/1, by vol.) and B: MTBE/methanol/water (90/9/1, by vol.).

Gradient conditions: equilibration conditions at 60 % B (v/v); 0 to 40 min linear gradient from 60% B to 100 % B (v/v); 40 to 45 min 100 % B (v/v). Standard curves correlating areas to concentration of lycopene and tetradehydrolycopene (TDHL) was used to determine the quantity of these carotenoids in the samples. HPLC-MS analysis was carried out on a HPLC coupled to a UV-detector and a mass spectrometer (LCMS-2010EV/dual ion source DUIS2010, Shimadzu, Germany), using identical conditions as described above.

For the semi-preparative isolation of TDHL, E. coli BW-EB pJF-crtI was cultivated in 400 ml minimal medium (3 g KH 2 PO 4 l -1 , 12 g K 2 HPO 4 l -1 , 5 g (NH 4 ) 2 SO 4 l -1 , 0.3 g MgSO 4 .7H 2 O l -1 , 0.015 g CaCl 2 .2H 2 O l -1 , 0.1 g NaCl l -1 , 5 g Glucose monohydrate l -1 , 15 ml FeSO 4 .7H 2 O /sodium citrate l -1 (from a solution of 7.5 g FeSO 4 .7H 2 O l -1 and 100 g sodium citrate l -1 ), 7.5 mg thiamine l -1 , 33 ml trace elements l -1 ) at 30°C. The culture was induced with 0.5 mM IPTG at OD 600 of 0.3. After 24 h, the cells were harvested by centrifugation, washed with cold water, and subsequently extracted twice with 20 ml acetone by vigorous shaking. Insoluble components of the extract were removed by centrifugation (20,000 x g). Acetone was removed under a stream of nitrogen and the residual material was dissolved in MTBE. TDHL was isolated by HPLC (several runs) using the same device as for analysis. The mobile phase was MTBE/methanol/water (90/9/1, by vol.) at 1.2 ml min -1 for 20 min isocratic; 200 µl as injection volume. TDHL fractions were collected, dried under a stream of nitrogen and subsequently dissolved in acetone. TDHL content in acetone solution was quantified by measuring the absorbance at 508 nm. Concentration of TDHL was determined using an extinction coefficient (5933 (g/ml) -1 cm -1 , [START_REF] Unrean | Rational design and construction of an efficient E. coli for production of diapolycopendioic acid[END_REF].

Results and discussion

To investigate the effect of single-and multi-copy expression of crtE, crtB, and crtI from P.

ananatis on lycopene biosynthesis in E. coli, strains with all different combinations of two chromosomal and one plasmid-encoded crt-gene were constructed. The expression of each gene is controlled by an IPTG-inducible tac-promoter. The chromosomal insertion of the heterologous expression cassettes were conducted by a method described recently [START_REF] Albermann | A simple and reliable method to conduct and monitor expression cassettes integration into the Escherichia coli chromosome[END_REF]). This method is based on the -Red-recombineering technique [START_REF] Datsenko | One-step inactivation of chromosomal genes in Escherichia coli K12 using PCR products[END_REF]Zhang et al. 1999). The method allows for a fast and accurate screening of the recombination event by differential MacConkey agar medium. The following strains were received after integration of two individual crt-expression cassettes into the chromosome of E. coli BW25113 lac + : E. coli BW-EB, E. coli BW-EI, E. coli BW-BI (table 1). To allow the biosynthesis of lycopene in these strains, each of the strains was transformed with the multi-copy expression vector pJF119N carrying the respective crt-gene, so that each strain carries the heterologous genes crtE, crtB, and crtI, resulting in E. coli BW-EB pJF119-crtI, E. coli BW-EI pJF119-crtB, and E. coli BW-BI pJF119-crtE, respectively.

These strains were cultivated in LB medium and the cultures were induced with different concentrations of IPTG (0 -1 mM). E. coli BW-Lyco, that contains all three lycopene biosynthesis genes on the chromosome, E. coli BW-Lyco pJF119N, and E. coli BW25113 lac + pJF119-crtEBI were used as strains for comparison.

The lycopene content per volume as well as per g cell dry wt are given in Figure 1. E. coli BW-Lyco, which expresses crtE, crtB, and crtI from individual chromosomal copies, had the highest lycopene formation (3.9±0.1 mg l -1 ; 1.65±0.12 mg g -1 cdw). The enhanced expression of dxs, idi, and ispA by E. coli BW-Lyco pJOE5559 resulted in the formation of 8.72±0.21 mg l -1 , 4.45±0.32 mg g -1 cdw lycopene showing that the enzyme activities of CrtE, CrtB, and CrtI expressed by single chromosomal gene copies is sufficient for the conversion of FPP to lycopene even under conditions with increased supply of isoprenoid precursors. The induction of these strains with IPTG had no significant effect either on growth or on lycopene formation. E. coli BW-Lyco pJF119N, carrying an additional lacI gene on the plasmid, showed an increase in lycopene formation after increasing the inducer concentration showing that the expression of the chromosomal lacI is not sufficient to repress the tac-promoter under the given conditions. A higher LacI concentration, as given in E. coli BW-Lyco pJF119N, led to an IPTG-inducible expression of the heterologous genes.

Expression of lycopene biosynthesis genes by an artificial, plasmid encoded crtEBI-operon (E. coli BW pCAS58) yielded 1.77±0.23 mg l -1 , 0.84±0.11 mg g -1 (cdw) lycopene without induction. The amount of lycopene decreased upon induction by IPTG. The plasmid expression of crtE by E. coli BW-BI pJF119-crtE under different inducer concentrations led to a similar amount of lycopene per cell (1.35±0.08 mg g -1 cdw) but the total biomass decreased due to the high expression of crtE. The higher lycopene content in E. coli BW-BI pJF119-crtE compared to E. coli BW-Lyco pJF119N in cultures without induction indicates that the CrtE catalyzed reaction is the most limiting step in the conversion of FPP to lycopene.

The crtB plasmid expression by E. coli BW-EI pJF119-crtB led to a decrease in growth as well as on lycopene formation. The CrtB product phytoene was not detected in the cell extract. Growth retardation may occur due to the interference of CrtB with the formation of prenylquinones in E. coli [START_REF] Neudert | Expression of an active phytoene synthase from Erwinia uredovora and biochemical properties of the enzyme[END_REF]), but the low biomass and lycopene amount per cell in the stationary phase (after 24 h) in induced cells (> 0.1 mM IPTG) suggests that the CrtB protein also being toxic or a protein burden to E. coli. This is also supported by the observation that E. coli BW-EI pJF119-crtB pJOE5559, which showed increased formation of FPP by the additional copies of dxs, idi, and ispA, had reduced growth under high crtB expression conditions (data not shown).

A further effect on the plasmid overexpression of a single crt-gene was observed by the expression of crtI. Cultivation of E. coli BW-BE pJF119-crtI yielded, beside lycopene, predominantly in the formation of 3,4,3´,4´-tetradehydrolycopene (TDHL) (Fig. 2), which was identified by LC-MS analysis (M + 532) and UV/visible spectroscopy (Fig. 2H). For the quantification of TDHL, E. coli BW-BE pJF119-crtI was cultivated in minimal medium and the extracted TDHL was isolated by semi-preparative HPLC (Fig. 2F). The concentration of TDHL in acetone was determined by photospectroscopy [extinction coefficient 5933 (g/100ml) -1 cm -1 , [START_REF] Unrean | Rational design and construction of an efficient E. coli for production of diapolycopendioic acid[END_REF]]. The amount of TDHL in LB cultures of E. coli BW-BE pJF119-crtI increased upon IPTG induction (Figs. 1 and2). The highest TDHL concentration was after induction with 0.5 mM IPTG (253±33 µg g -1 cdw, 59% of all carotenoids).

Induction with 1 mM IPTG gave the highest percentage of TDHL with 80% (169±27 µg g -1 cdw). Extracts of cultures with a high TDHL content as well as the purified TDHL samples also reveal the existence of two further isomeric forms of TDHL (M + 532) as indicated in Fig.

2. The formation of TDHL was not observed in any of the other tested lycopene producing strains, in which crtI is expressed from single copy or by the plasmid encoded gene cluster (pCAS58).

For the CrtI-type phytoene-desaturase from P. ananatis and other organisms, the general ability to carry out up to six desaturation reactions on phytoene has been shown [START_REF] Linden | Functional complementation in Escherichia coli of different phytoene desaturase genes and analysis of accumulated carotenes[END_REF][START_REF] Sandmann | Evolution of carotene desaturation: the complication of a simple pathway[END_REF]. Additionally, the ratio of substrate and desaturase can further define the degree of phytoene desaturation [START_REF] Stickforth | Kinetic variations determine the product pattern of phytoene desaturase from Rubrivivax gelatinosus[END_REF]). Sandmann and coworkers used a two plasmid system for the differentiated expression of crtEB and crtI in E.

coli, which resulted in TDHL at approx. >10% of all carotenoids. Here, the reported combination of single-copy (crtE, crtB) and multi-copy (crtI) expression led to a TDHL portion of up to 80% of the E. coli synthesized carotenoids which verifies that the kinetics of CrtI from P. ananatis can determine the degree of desaturation. By the use of directed evolution methods a CrtI-mutant was generated that led, compared to the wild type enzyme, to a partly conversion of lycopene to TDHL in E. coli [START_REF] Schmidt-Dannert | Molecular breeding of carotenoid biosynthetic pathways[END_REF]. Due to the lack of biochemical data it is not yet clarified if the mutein has an increased catalytic activity or if the amino acid exchanges lead to a higher protein overproduction or stability that caused the formation of TDHL.

This report demonstrates that the expression level of heterologous biosynthesis pathways may have a significant impact on the product yield as well as on product specificity. The expression of single chromosomal copies of either crtE, crtB or crtI leads to an about 90% higher yield of lycopene compared to the plasmid expressed gene cluster. E. coli BW-BE pJF119-crtI produced TDHL as the predominant carotenoid, due to the high expression of crtI, which opens the opportunity for a preparative synthesis of this rare carotenoid.

In conclusion, this report demonstrates that an efficient heterologous in vivo biosynthesis of carotenoid compounds requires an adapted expression of the heterologous genes to avoid metabolic burden effects and unspecific side reactions.

Legends to the figures

Figure 1 :

 1 Figure 1: Formation of lycopene and tetradehydrolycopene (TDHL) by recombinant E. coli

Figure 2 :

 2 Figure 2: HPLC-analysis of carotenoids in extracts from recombinant E. coli strains after

  Figure 1

Acknowledgment

The author is grateful to Holger Beuttler (Universität Stuttgart) for LC-MS measurements and for providing carotenoid standards. I thank Josef Altenbuchner (Universität Stuttgart) for providing plasmid pJOE5559. This work was supported by the Deutsche Forschungsgemeinschaft (SFB706, TP B3).


Martin Schoen
email: martin.schoen@tu-berlin.de

Gerrit Günther

Capillary condensation in deformable mesopores: Wetting versus nanomechanics

Keywords: Phase transition, Capillary condensation, Sorption strain, Monte Carlo simulation

published or not. The documents may come

Capillary condensation in deformable mesopores:

Wetting versus nanomechanics.

Nanoconfined fluids have been under intense investigation for quite some time because of their unique and fascinating properties [1][START_REF] Schoen | Nanoconfined fluids. Soft matter between two and three dimensions[END_REF][START_REF] Schoen | [END_REF][4]. These properties are caused by a competition between several length scales such as the range of the intermolecular interaction potentials, the characteristic dimensions of the nanoscopic container to which the fluid is confined, or any chemical or geometrical patterns with which the inner surfaces of these containers may be endowed.

A particularly interesting realization of nanoconfined soft-matter phases are fluids adsorbed to mesoscopic porous solids. These systems have received a lot of attention because they play a key role in many important applications or processes such as nanoreactors in catalysis [5], filtering processes [6], or the purification of tiny amounts of multi-component mixtures [7]. Most of these applications are possible only because of the large internal surface of mesoporous materials. The broad variety of practical applications has spurred the development of many different mesoporous materials such as glasses and ceramics [8] or various carbon materials [9]. Very recently, even metallic systems have been synthesized [10]. A broad variety of adsorbates ranging from classical polar and nonpolar fluids [11,12] to quantum fluids [13] have been investigated. Particularly prominent examples both from an experimental and from a theoretical perspective are mesoporous silicas such as MCM-41 [14] or SBA-15 [15]. This is because these mesoporous solids consist of regular hexagonal arrays of parallel, unconnected cylindrical pores with typical pore widths in the range of 2-10 nm. These materials are particularly fascinating because the pore-size distribution is narrow and because individual pores are accurate to shape.

Compared with a single solid surface, confinement to spaces of nanoscopic dimensions has important ramifications for adsorption and desorption phenomena (which we shall subsequently subsume under the term "sorption") and therefore for the phase behavior of such nanoconfined fluids [1][START_REF] Schoen | Nanoconfined fluids. Soft matter between two and three dimensions[END_REF][START_REF] Schoen | [END_REF][4]. From a broad spectrum of theoretical approaches ranging from density functional theory over integral-equation techniques to computer simulations a rather detailed picture of phase transitions in nanoconfinement has emerged over the years [1][START_REF] Schoen | Nanoconfined fluids. Soft matter between two and three dimensions[END_REF][START_REF] Schoen | [END_REF][4].

Traditionally, in almost all these studies the confining solid material has been perceived as a representation of a temporally static external field that may only vary spatially. An example of this kind of a situation is a fluid confined by nonplanar or chemically patterned substrates [START_REF] Schoen | [END_REF] where neither the geometry nor the shape of the chemical pattern varies with time. In other words, the underlying assumption in the overwhelming number of previous works has been that the mesoporous solid does not "respond" to whatever changes may occur in the thermophysical properties of the confined fluid phase. However, it has recently been demonstrated by us that deformability of the mesoporous matrix may have important consequences for the phase behavior of the confined fluid [16]. Consequently, we deviate from most earlier work by allowing the external field to vary during the course of the simulation (i.e., with time). This permits us to consider confining substrates that may to a certain extent deform in response to changes in the thermodynamic state of the confined fluid.

Our approach is motivated by the experimental observation that mesoporous materials can by no means be considered to be completely rigid. Mechanical deformation has been observed, for example in mesoporous silicon, where the lattice constant is changed in comparison to bulk silicon upon fluid sorption and condensation [17]. Other examples are mesoporous glasses that were found to deform elastically during capillary condensation [12,18]. Moreover, mechanical deformation of the solid support has been observed in mesoporous metals as a consequence of surface charging or fluid wetting and condensation [19,20]. This shows that the enormous amount of surface per unit volume makes a significant contribution to the total energy of the system in mesoporous materials. By using a single-pore model it could be demonstrated by Günther et al. that pore deformation has a marked impact on the location of phase boundaries between coexisting gas-and liquid-like phases of the confined fluid [18]. Prass et al. recently showed that elastic properties of the mesoporous solid may be determined quantitatively from in-situ small-angle X-ray diffraction experiments during the sorption process [12].

One of the interesting consequences of this condensation-induced deformation is that it constitutes a macroscopic signature of a microscopic effect related to the nature of the fluid phase in the mesopores. In this sense, macroscopic deformation of the solid support may be used to sense fluid condensation or, conversely, the support can be seen as being actuated by changes in the pores at the molecular level. Recently, nanoporous metals have been discussed as actuators driven by various signals on the internal surface, such as charge or fluid condensation [19,20]. Here, we discuss similar effects in glassy silica, which, compared to metals, have the advantage of comparatively low reactivity, as well as higher stiffness and lack of plastic deformation. Our focus will be on strain isotherms, that is the strain "experienced" by the solid matrix as the pore fills with fluid. At low pressures, at which only a thin fluid film is adsorbed onto the solid surfaces, strain isotherms depend on wetting characteristics of the solid material. However, once the pore is completely filled with fluid material strain isotherms reflect the elasticity of the confining solid.

Our paper is organized as follows. In Sec. II we give a brief account of relevant experimental techniques and results. In Sec. III we introduce our model system. Relevant methodological and theoretical aspects of the Monte Carlo simulations, on which this work is based, are summarized in Sec. IV. Section V is given to a presentation of the results obtained which are discussed and put into perspective in the concluding Sec. VI.

II. EXPERIMENTAL BACKGROUND

To make contact with a specific experimental situation we focus on sorption experiments [21] employing ordered mesoporous silica of the type MCM-41 [14]. These materials consist of grains of roughly 1 µm size, containing cylindrical nanopores on a well ordered twodimensional hexagonal pore lattice with pore diameter of the order of 3-4 nm and narrow size distributions. Typical lattice parameters, pore sizes, as well as the pore volume fractions of these materials are listed in Table 2.1 of Ref. 21. For the sake of concreteness we focus only on experiments using pentane as the adsorbate. During the sorption process in-situ X-ray diffraction measurements are performed using synchrotron radiation [22]. The experimental setup is described elsewhere [11,23] and consists essentially of a thermostatted sample chamber with X-ray transparent windows, which is connected to a fluid reservoir via a gas dosing system. The gas pressure in the sample chamber is changed slowly but continuously while the X-ray powder patterns are recorded. This provides datasets of the diffracted intensity as a function of the relative gas pressure P/P 0 and the magnitude of the scattering vector q. Upon filling the contrast between the solid matrix and the interior of the pores changes. Therefore, a decrease in diffracted intensity with increasing P/P 0 can be attributed to enhanced adsorption of fluid material by the mesoporous matrix.

The region of capillary condensation is easily recognized as a sharp drop in intensity from the diffraction data due to the contrast change upon pore filling [see Fig. a (P ) -a 0 a 0 = q hk (0) -q hk (P )

q hk (P ) (2.1)
as a quantitative measure of substrate deformation during the sorption process. In Eq. (2.1), a (P ) and a 0 are the lattice constants of the hexagonal array of cylindrical pores filled with fluid at pressure P and of the empty mesoporous matrix, respectively. Similarly, q hk (P ) and q hk (0) are the magnitudes of the scattering vector of the respective Bragg peak hk at bulk pressure P and of the empty mesoporous matrix. The far right side of Eq. (2.1) obtains because q hk ∝ 1/a. As the relative shifts were identical for all reflection orders hk, it has been concluded that the associated lattice strain ε is isotropic within the plane perpendicular to the pore axis [12]. This implies that during the sorption process the radius of each pore changes isotropically such that its cylindrical shape remains unaffected as well as its length.

Hence, the deformation of the sample may be discussed appropriately in terms of the scalar quantity ε defined in Eq. ( The parallel theoretical analysis is based upon a couple of assumptions. First, we assume the filling of these pores to be essentially controlled by the (slightly different) widths of individual pores. Pores of different widths are distributed randomly across the macroscopic grains of which the mesoporous solid consists. If the pressure P in a bulk reservoir is larger than that pressure at which capillary condensation occurs in an individual pore of a certain width all pores up to the same pore diameter would completely and spontaneously be filled with a liquid-like phase. Like the empty pores of the pristine sample these now liquid-filled pores are also distributed at random across the mesoporous grains.

It has been suggested recently [24][25][26] that upon filling with liquid the subsequent deformation of an individual pore might have an impact on neighboring pores as well and this would cause a transmission of sorption-induced strain over a certain distance in the mesoporous solid. This, in turn, could affect the pore size distribution such that pores of certain sizes are no longer randomly distributed across the sample. Instead, a sorption-strain driven spatial correlation of pore sizes could exist which then would lead to some sort of clustering of liquid-filled pores in specific regions of the mesoporous solid. This notion prompted Grosman and Ortega [25] to propose a long-range interaction between individual pores through which local elastic strains are transmitted through the mesoporous solid. These authors base their proposition on detailed analyses of sorption isotherms where the surmised correlation effects can be ascertained only rather indirectly. However, very recent neutron scattering experiments on SBA-15 silica contradict the clustering phenomenon [27]. The neutron scattering data are fully consistent with the assumption that the spatial distribution of pores of different sizes remains random despite the sorption-induced deformation of individual pores in the sample. This observation permits us to theoretically describe the sorption process essentially in terms of a single isolated pore.

Moreover, as long as very narrow pores of typical widths of only a few nanometers are concerned, the precise geometry of the pore is subdominant. For example, capillary condensation, which constitutes a gas-to-liquid phase transition in nanoporous materials, occurs in slit-pores and in cylindrical pores irrespective of pore geometry. To establish contact between theory and experiment in this manuscript we estimate both the diameter σ of a hypothetical pentane molecule in a dense liquid phase and a realistic distance s z0 of the Introducing next the concept of a degree of confinement [28] permits us to match the cylindrical pores of the experimental systems to a model slit-pore in the Monte Carlo simulations where both have the same ratio of pore volume to surface area. Applying this concept we obtain a pore width of s ′ z0 = 4.8σ for the empty model slit-pore based upon experimental data for MCM-41 (16) (see Table 2.1 of Ref. 21) which we adopt for sake of concreteness and to enable a direct comparison between experimental and theoretical data discussed in this work. Finally, noticing that both wall atoms and fluid molecules are assumed to be of the same size and on account of repulsive forces between them there is a distance of about σ at each of the solid substrates that is inaccessible to fluid molecules. Thus, the correct width of our slit pores (in units of σ) is given by s z0 = s ′ z0 + 2 = 6.8 which is the value used in all our Monte Carlo simulations throughout this work. In a similar spirit we wish to abstract from details of the chemical structure of the adsorbate to keep the complexity of any model system to the absolute minimum. For a study of gas-liquid phase transition the only necessary ingredient are attractive van der Waals forces between the molecular constituents. This is reflected by the fact that even a simple mean-field type of approach is capable of capturing key aspects of the effect of nanoconfinement on gas-liquid phase transitions in both pure fluids and binary mixtures [START_REF] Schoen | Nanoconfined fluids. Soft matter between two and three dimensions[END_REF].

III. MODEL SYSTEM

We base the theoretical work of the present study on a model consisting of sphericallysymmetric particles (i.e., wall atoms and confined-fluid molecules) which we assume to interact with each other in a pairwise additive fashion. In view of the experimental situation summarized in Sec. II we base our numerical analysis on a model of maximum simplicity retaining, however, key features of the experimental situation. Specifically, we express the fluid-fluid contribution to the total configurational energy as where r ij = |r ir j | is the distance between molecules i and j located at r i and r j , respectively, R ≡ (r 1 , r 2 , . . . , r N ) represents the configuration of fluid molecules,

U ff (R) = 1 2 N i=1 N j=1 =i u (r ij ) ( 3
u (r) =    u sh (r) , r ≤ r c 0, r > r c (3.2)
where

u sh (r) = u LJ (r) -u LJ (r c ) - du LJ (r) dr r=rc (r -r c ) (3.3)
is a so-called shifted-force potential. Unlike the full Lennard-Jones (LJ) potential

u LJ (r) = 4ǫ σ r 12 - σ r 6 (3.4)
u sh (r) vanishes continuously together with its first derivative (i.e., the intermolecular force) at the radius r c of a cutoff sphere centered on r i . Throughout this work we use r c = 3.5σ for all interactions. Because of its definition and unlike u LJ (r), u sh (r) is explicitly shortrange which is advantageous in the Monte Carlo simulations on which this work is based [START_REF] Schoen | Nanoconfined fluids. Soft matter between two and three dimensions[END_REF]29]. In Eq. (3.4), σ is the "diameter" of a fluid molecule and ǫ determines the strength of intermolecular interactions in the usual manner where we henceforth identify σ with the diameter of a hypothetical spherical pentane molecule.

As in our previous work [16,18,21,30] fluid molecules are confined by two planar solid substrates. The substrates consist of individual solid atoms whose equilibrium positions are distributed according to the (100) plane of the face-centred cubic (fcc) lattice where ℓ/σ = 3 √ 4 is the lattice constant. Hence, the substrates are characterized by an areal density ρ s = 2/ℓ 2 . Equilibrium positions of substrate atoms are so arranged that they are distributed across an area A 0 in the x-y plane. In this case the distance between each pair in upper and lower substrate is s z0 . Because the substrates are atomically structured there is a fluid-solid (fs) contribution to the total configurational energy. It can be cast as

U fs (R, R s ) = α 2 k=1 N i=1 Ns j=1 u(|r i -r [k] j |) (3.5)
where N s = 2n 2 is the total number of substrate atoms located in wall k and n is an integer specifying the number of unit cells of the fcc (100) plane. In Eq. (3.5), R s = (r

[1] 1 , . . . , r [1]
Ns , r

[2] 1 , . . . , r [2]
Ns ) represents the configuration of substrate atoms in both substrates identified through the superscripts. The potential u in Eq. (3.5) is given in Eq. (3.2). The Because we are concerned with sorption-induced deformations of the substrates and the impact of these deformations on the sorption process the substrates themselves need to be thermally coupled to the confined fluid phase. This way they can respond to whatever change in the thermodynamic state of the confined fluid may occur. Thermal coupling requires the substrate atoms to be able to depart from their equilibrium lattice sites {r

[k]
i0 } to a certain extent under the action of both neighboring solid atoms and fluid molecules. Thermal coupling causes a deformation of the solid substrates such that a solid-solid contribution to the total configurational energy arises. It may be cast here as

U ss = 2 k=1 α 2 Ns-1 i=1 Ns j=i+1 u(|r [k] i -r [k] j |) + κ Ns i=1 |r [k] i -r [k] i0 | 2 (3.6)
In this expression u is again given by Eq. (3.2). The first term on the right side of Eq. (3.6) is necessary because substrate atoms must not approach each other arbitrarily (and therefore unrealistically) close. To prevent the substrates from "melting" we bind each solid atom to its equilibrium lattice site by introducing a harmonic potential [see second term in brackets in Eq. (3.6)] where κ > 0 determines the binding strength (i.e., the "stiffness" of a harmonic spring). In other words, κ is a measure of rigidity of the substrates. For values of κ used below we utilized a modified Lindemann criterion to ensure that over the temperature ranges studied the solid substrates do in fact not melt [30]. Hence, in the system of interest the total configurational energy may be expressed as [see Eqs.

U (R, R s ; κ) = U ff (R) + U fs (R, R s ) + U ss (R s ; κ) (3.7)
and depends on R, R s , and κ as a parameter. Setting κ ≥ 30 and s z = 6.8σ guarantees that pairs of solid atoms bound to different substrate planes do not interact with one another.

IV. THEORETICAL AND NUMERICAL ASPECTS

As we pointed out in Sec. II experimentally the adsorbate is in contact with a bulk reservoir of the same fluid. Ideally the two are in thermodynamic equilibrium during the sorption process. Assuming the reservoir to be sufficiently large such that its density does not change appreciably during the sorption process, which is guaranteed by the experimental setup, suggests the grand potential Ω as the natural thermodynamic potential for the subsequent analysis of strain isotherms. As we already demonstrated elsewhere [21] the exact differential of Ω may be cast as dΩ (T, µ, A, s z ) = -SdT -Ndµ -P s z0 dA -

P zz A 0 ds z (4.1)
where T is temperature, S is entropy, N is the number of molecules of which the adsorbate consists, and µ is its chemical potential. The last two terms in Eq. (4.1) represent the mechanical work associated with components P and P zz of the pressure tensor P acting in directions parallel and perpendicular to the plane of the solid substrates, respectively.

Because no shear forces are acting on the confined fluid, P is diagonal. Notice also that on account of nanoconfinement P = P zz in general whereas

lim sz→∞ P = lim sz→∞ P zz = P (4.2)
Because of thermal coupling between adsorbate and adsorptive both the fluid-solid interfacial area A and the substrate distance s z may in principle deviate from the equilibrium values A 0 and s z0 [21].

To link the above phenomenological treatment to a molecular level of description we employ conventional concepts of statistical thermodynamics. To that end

Ω = -k B T ln Ξ (4.3) is the key expression where k B is Boltzmann's constant, Ξ = ∞ N =0 exp (βµN) Q (4.4)
is the grand canonical partition function, and β ≡ 1/k B T . In the classical limit [START_REF] Schoen | Nanoconfined fluids. Soft matter between two and three dimensions[END_REF], with which we are exclusively concerned in this work, the canonical-ensemble partition function may be cast as

Q = Z Λ 3(N +2Ns) N! (4.5)
where

Λ = h √ 2πmk B T (4.6)
is the thermal de Broglie wavelength, h is Planck's constant, and m is the mass of a fluid molecule. We note that all thermodynamic quantities of interest can be expressed as partial Finally, for our system of interest the configuration integral in Eq. (4.5) is then given by

Z (T, N, N s , A, s z ) = dR dR s exp [-βU (R, R s ; κ)] (4.7)
where the dependence on A and s z is buried in the limits of integration with respect to dard fashion [START_REF] Schoen | Nanoconfined fluids. Soft matter between two and three dimensions[END_REF]31]. Because all interaction potentials are short-ranged we employ periodic boundary conditions at the planes x = ±s 0 /2 and y = ±s 0 /2. To save computer time a

Verlet neighborlist is combined with a link-cell list [START_REF] Allen | Computer simulation of liquids[END_REF].

V. RESULTS

A. Strain isotherms

As we have argued in Sec. II the specific experimental conditions permit us to describe the sorption process essentially in terms of a single pore. However, this poses an immediate problem to a direct comparison with the experimental results which were obtained for a regular array of disconnected individual pores. On account of its regularity the mechanical state of the latter may be characterized by a sorption-induced lattice strain ε defined in Eq. (2.1). Obviously, for the theoretical model a meaningful definition of ε is impossible because we are dealing with a single pore rather than with an ordered array of pores. However, a change of a in the experimental system suggests that upon sorption of fluid material each individual pore of the entire matrix changes its width so that ε will be proportional to that change. However, the experimental observation of in-plane isotropy of the lattice strain permits us to discuss substrate deformation in terms of a single scalar parameter, too (see Sec II). To that end we define an effective pore width s eff z ≡ z [2] -z [1] (5.1)

where

z [k] ≡ 1 N s Ns i=1 z [k] i (5.2)
and . . . denotes an average in the grand canonical ensemble. With these definitions we introduce the pore strain

ξ = s eff z -s z0 s z0 (5.3)
as a quantitative measure of deformation of the slit-pore. The above considerations immediately suggest that ε ∝ ξ thereby enabling a qualitative comparison between experiment and simulation (see also Sec. VI). bulk reservoir with which the confined fluid phase is in equilibrium where P + 0 is the (bulk) pressure at which capillary condensation occurs. The overall agreement between theoretical and experimental data is very gratifying especially in view of the relative crudeness of the theoretical model. The reader should note, however, that on account of this crudeness of the theoretical model P + 0 will differ between the experimental and theoretical data sets. However, apart from this restriction the plots in Fig. 5.1 reveal a remarkable, semi-quantitative agreement of the two sets of data. We therefore conclude that with proper calibration the theoretical model is sufficiently realistic to reveal molecular details about the relation between substrate deformation and the sorption process in mesoporous materials. That the discontinuity at P/P + 0 = 1.00 reflects capillary condensation is inferred from a discontinuous change in the sorption isotherm also plotted in Fig. 5.1. In our Monte Carlo simulations we have direct access to the pressure (or chemical potential) at phase coexistence and the associated mean densities of the fluid in the pore through the procedure described in Ref. 16.

Note also that the sorption isotherm plotted in Fig. 5.1 does not exhibit hysteresis unlike its experimental counterpart would. Hysteresis is due to the existence of metastable thermodynamic states of considerable lifetime. However, within the framework of an equilibrium simulation method such as Monte Carlo metastability must be regarded an artifact with no real physical meaning as was pointed out quite some time ago by Schoen et al. [START_REF] Schoen | [END_REF].

That both lattice strain ε and pore strain ξ drop to negative values at capillary condensation can be understood if one keeps in mind that condensation of fluid phases depends crucially on attractive intermolecular interactions. Hence, at capillary condensation the solid atoms in our slit-pore model are subject to an attractive field exerted by the condensed liquid-like confined fluid which consequently pulls them inward. This causes s eff z < s z because of our definition of the effective pore width given in Eq. (5.1). Therefore, at capillary condensation ξ < 0 as one can easily verify from Eq. (5.3). Another macroscopically motivated argument for the drop of the lattice strain to negative values was used in Ref. 12.

It is based on the Laplace pressure between the liquid-and the gas phase separated by a meniscus. As this pressure is negative, a compressive deformation of the pore walls is expected at capillary condensation.

Data plotted in Fig. 5.1 are obtained for α = 1.20 [see Eq. (3.5)], that is for pore walls that are wet by the adsorbate. To investigate the impact of different wetting scenarios on the shape of the strain isotherm in more detail we plot in Fig. 5.2 data for different strengths of the fluid-substrate interaction by varying α in Eq. (3.5). Focusing on low-pressure portions of the isotherms first we see from Fig. 5.2(a) that ξ increases slightly with P up to that pressure at which capillary condensation occurs (P/P 0 ≃ 0.70). Over a comparable range of pressures the plot in Fig. 5.2(b) indicates that ξ ≈ const. if one enhances the wettability of the substrate from α = 1.0 to 1.5. For substrates that are even better wet by the fluid phase the plot in Fig. 5.2(c) reveals that even at low bulk pressures the slope of the sorption strain is negative indicating that in this case the fluid-substrate attraction is strong enough to slightly pull the solid atoms inward even before the pore begins to imbibe fluid material appreciably.

These phenomena can be correlated with variations of P zz . There exist two different ways of deriving molecular expressions for P zz . Traditionally, we refer to those as the "force" and

P zz = sgn ( F [1] z ) 2A 0 2 k=1 F [k] z (5.4)
where [see Eq. (3.5), Ref. 31]

F [k] z = - N i=1 Ns j=1 ∂ ∂z [k] ij u(r [k] ij ) (5.5) r [k] ij = r i -r [k]
j , and sgn x = ±1 if x ≷ 0, respectively. In a similar fashion the "virial" where

P id zz = N k B T A 0 s z (5.7a) P pot zz = P ff zz + P fs zz = - 1 2 1 A 0 s z N i=1 N j=1 u ′ (r ij ) r ij ( r ij • e z ) 2 - 1 A 0 s z 2 k=1 N i=1 Ns j=1 u ′ (r [k] ij )r [k] ij r [k] ij • e z r [k] ij + δ r [k] ij • e z (5.7b)
where notation " ..." is used to refer to unit vectors. In particular e z is a unit vector pointing along the z-axis of the Cartesian coordinate system. In Eq. (5.7b) + F

δr [k] ij ≡ r [k] i -r [k] i0 -r [k] j -r [k] j0 (
[2] z = 0. Entries in Table 5.1 illustrate the agreement between "force" and "virial" routes.

Plots in Fig. 5.2 illustrate the correlation between P zz and ξ. In particular, the marked drop of ξ at pore filling is accompanied by a substantial decrease of P zz . Once the pore is filled with a liquid-like phase further increase of the bulk pressure causes both P zz and ξ to increase, too. Even at bulk pressures below pore filling the initial slope of ξ seems correlated with the variation of P zz even though this correlation is more difficult to see on account of the small values of P zz in this regime. Nevertheless, the dependence of ξ on α in Fig. 5.2

prior to pore filling indicates that in the low-pressure regime the pore strain depends on wetting characteristics of the substrate. To see this we analyze the local density However, once the pore is filled with a liquid-like phase variations of the wetting characteristics of the substrate do not matter as one might have anticipated intuitively. This is illustrated in Fig. 5.4 where the pore strain is plotted as a function of the bulk pressure for various values of α. As one can see from the plot all data sets collapse onto a straight line irrespective of α. The reader should appreciate that in Fig. 5.4 data are presented over a range from moderately drying (α = 0.75) to strongly wetting (α = 1.50) substrates. This observation is in line with experimental results where for the same mesoporous solid ε turns out to be independent of the chemical nature of the confined fluid [12]. If, on the other hand, the deformability of the substrates is altered the pore strain changes markedly. This can be seen from Fig. 5.5 where ξ still depends linearly on ln(P/P 0 ) similar to the plot in Fig. 5.5 but the slope of the straight lines increases with decreasing stiffness of the harmonic binding potential in Eq. (3.6). Similar effects have been seen experimentally for various MCM materials (see Fig. 9 of Ref. 21). Depending on conditions of the synthesis of the mesoporous solids materials with different pore widths D and lattice parameters a 0 can be realized. If a 0 varies the thickness of the solid matrix between neighboring pores changes, too. Therefore, materials with larger a 0 are expected to be stiffer. Plots in Fig. 9 of Ref. 21 reveal that the slope of lattice strain ε versus ln(P/P 0 ) decreases with increasing a 0 similar to the variation of pore strain ξ with κ in the theoretical model (see Fig. 5.5).

ρ(z) ≡ N i=1 δ (z -z i ) = N (z) A 0 δ z (5.
Once again this observation supports our earlier notion that ε ∝ ξ and provides additional support for the validity of our single-pore model.

B. Thermomechanical analysis and quasi-Kelvin equation

To rationalize the logarithmic dependence of ξ on P we start from Eq. (4.1). For the subsequent discussion it will prove useful to introduce the Gibbsian potential G ≡ Ω+A 0 s z P zz as the relevant thermodynamic potential through a Legendre transformation of Ω such that dG = -SdT -Ndµ -P s z0 dA + A 0 s z dP zz (5.10) follows. Under conditions of fixed T , µ, and P zz , G is approximately a homogeneous function of degree one in A as we explained elsewhere [21]. Hence, we may apply Euler's theorem which gives (see, for instance, Appendix A. To proceed we perform a gedankenexperiment in which we decompose the entire confined fluid into a lamella of thickness s z and finite extent A in the x-y plane. Henceforth, the lamella is perceived as "the system" in a thermodynamic sense. Consequently, the remainder of the confined fluid plus the bulk are taken as "the environment". The planes x = ±s x /2 and y = ±s y /2 separating the system from the environment may be viewed as imaginary and immaterial pistons through which the system can exchange mechanical work with its environment. It is then conceivable that upon varying the bulk pressure P one could readjust the pistons such that P is held constant. The mechanical work done on the system would then be completely transformed into a change of the lamella's thickness s z . Therefore, under the conditions of the gedankenexperiment we may drop the term As z0 dP in Eq. (5.12).

Notice also that the gedankenexperiment is fully consistent with the experimental observation of in-plane isotropy of the lattice strain which prevents pores from changing their length as the pressure of the bulk reservoir varies.

Hence, under conditions of fixed T and P we may rearrange terms and integrate Eq. (5.12) to obtain µ l [P zz (P 0 ) , T ] = µ l [P zz (P ) , T ] + 1 ρ l [P zz (P 0 ) -P zz (P )] (

where superscript "l" is used to emphasize that we focus solely on liquid-like thermodynamic states of the confined fluid, that is a situation in which the pore is completely filled with fluid material. We take this fluid to be incompressible such that its density ρ l ≡ N l /A 0 s z ≃ const.

Notation P zz (P 0 ) refers to the normal component of P at bulk saturation, that is at the bulk pressure P 0 at which the sorption experiment naturally ceases at a any given T . Obviously, this notation already takes notice of the fact that experimentally the confined fluid is in thermodynamic equilibrium with a bulk-gas reservoir maintained at some pressure P and temperature T . Note that for Eq. (5.13) to make sense, P ≤ P 0 must hold, too.

Under conditions of both experiment and simulation the confined liquid-like fluid is in thermodynamic equilibrium with bulk gas. Therefore, we may write Finally, to rationalize the plots in Fig. 5.5 we notice that under realistic conditions and for the present class of materials the substrate deformation is rather minute and typically of the order of 10 -4 . For such small deformations it seems reasonable to assume that the substrate responds to the confined fluid as a Hookean solid for which we may write P zz (P ) = Mξ (P ) (5.16) where we call M the "pore-load modulus".

≡ ρ l k B T M ∝ 1 κ (5.18)
under isothermal conditions. The plot in Fig. 5.6 confirms the linear dependence of m on 1/κ quite nicely. In retrospect, this indicates that concepts borrowed from phenomenological thermodynamics of macroscopic systems in combination with Hooke's Law for elastic solids provide an understanding of nanomechanical properties of mesoporous solids. The apparent applicability of these concepts demonstrated above provides a novel route for an experimental determination of nanomechanical properties of mesoporous solids from the response of the porous matrix to a liquid-like adsorbate.

VI. SUMMARY AND CONCLUSIONS

In this paper we discuss pore-strain isotherms computed in grand canonical ensemble Monte Carlo simulations of a "simple" Lennard-Jones fluid confined to a nanoscopic slitpore. In the simulations we employ a single slit-pore model. The walls of this slit-pore are composed of one (explicitly treated) layer of solid atoms bound to their equilibrium lattice sites through a harmonic binding potential of variable stiffness. The binding potential serves as a means to prevent the layer of solid atoms from melting and also mimics in a continuum sense the remainder of the solid matrix behind the first layer of atoms forming its surface.

Hence, solid atoms may depart from their equilibrium lattice sites because of their own kinetic energy, their interaction with neighboring solid atoms, and under the influence of their interaction with adsorbate molecules. Because solid atoms may move slightly during the course of the simulation a pore strain ξ may be calculated from an ensemble average of positions of the solid atoms. By tuning parameters of the model a semi-quantitative agreement between the measured lattice strain ε and ξ over the entire pressure range is obtained. This indicates that the somewhat simplistic model employed in the Monte Carlo simulations is sufficiently realistic to capture the essential physics governing the experimental results.

We focus primarily on branches of the strain isotherm before and after pore filling. Prior to pore filling the initial slope of the strain isotherm depends on the strength of the fluidsolid attraction. If this interaction is sufficiently weak the pore initially expands whereas contraction is observed if the fluid-substrate interaction strength exceeds a certain threshold.

These observations suggest that initially (i.e., at low bulk pressure) strain isotherms are dominated by wetting phenomena. This interpretation is further corroborated by plots of the local density showing that in the pressure regime considered the confined fluid consists of a thin film that is more strongly adsorbed by the solid material if the fluid-solid interaction is stronger.

A qualitatively different dependence of the strain isotherms is observed once the pore is completely filled with a liquid-like phase. In this regime the pore strain always increases with bulk pressure and exhibits a linear dependence on ln (P/P 0 ). We rationalize this linear relationship through a simple thermodynamic analysis which leads to an equation that permits one to deduce elastic properties of the porous solid (namely, the pore-load modulus). Experimentally, it is observed that the pore-load modulus varies as the pore volume fraction of the mesoporous solid is changed. However, the pore-load modulus turns out to be independent of the chemical nature of the confined fluid phase [12]. Similar observations are made in this work where we demonstrate that the high-pressure part of the strain isotherms is independent of the strength of the fluid-substrate interaction. In the theoretical model a change of the stiffness of the harmonic binding potential accounts for a change of the mesoporous material similar to that in the corresponding experimental system. In agreement with the experimental findings [21] it is observed in the Monte Carlo simulations that the linear relationship between ξ and ln (P/P 0 ) is preserved whereas the slope changes with varying stiffness of the confining solid material.

However, regardless of whether thermodynamic states prior or after pore filling are concerned the form of the strain isotherm appears to be correlated with variations of the normal component P zz of the pressure tensor P. Intuitively, one would expect such a correlation which indicates that variations of P zz with increasing bulk pressure P are the cause of pore deformation. The dependence of P zz on the bulk pressure P is equivalent to the dependence of the Laplace pressure Π of a liquid droplet (or a gas bubble) on P as predicted by the celebrated Kelvin equation [START_REF] Rowlinson | Molecular Theory of Capillarity[END_REF]. However, the Kelvin equation is usually derived by considering the coexistence of gas and liquid phases along an (arbitrarily) curved interface of some mean curvature. This thermodynamic analysis is combined with force-balance considerations at the gas-liquid-solid interface of a drop sitting on a substrate and in equilibrium with surrounding gas. The reader should note that the above derivation of a quasi-Kelvin expression given in Eq. (5.15) does not invoke assumptions about the specific geometrical properties of or forces acting on an interface between coexisting phases. The isomorphism between Kelvin's equation and Eq. (5.15) is merely a consequence of the assumption of thermodynamic equilibrium between a liquid and a gas phase as stated in Eqs. (5.14) where the latter is perceived as ideal. In textbook derivations of Kelvin's equation [START_REF] Rowlinson | Molecular Theory of Capillarity[END_REF] the interpretation of P and P 0 would be that of the gas pressure at saturation at a curved and at a planar interface, respectively.

Thus, the reader should appreciate the different physical situations depicted by Eq. (5.15) as opposed to that to which Kelvin's equation applies. Nonetheless, because Eq. (5.15) is identical in form with the classical Kelvin equation [except for the additive term P zz (P 0 )],

the use of such a thermodynamic expression together with a simple mechanical picture is useful to understand nanomechanical properties upon fluid condensation, as was shown in Ref. 12. The simple Kelvin equation approach was used to derive a pore-load modulus for materials with pore diameters as small as 3nm. For such small pores, the pore filling mechanism should not proceed via liquid-bridge formation (i.e., formation of meniscii), but rather via an axial filling mode [START_REF] Gruenberg | [END_REF]. However, because Eq. (5.15) is functionally equivalent to the classical Kelvin equation, but is not restricted to a specific filling mode via formation of a meniscus, the use of this phenomenological approach should not break down even for very small pores and should therefore remain extremely useful to understand and quantify nanomechanical properties upon fluid condensation.

However, very recently it has been suggested that for nanoporous metals surface-stress effects may outweigh the (Laplace-) pressure related phenomena discussed within the scope of the present model [36]. However, it remains to be demonstrated beyond doubt that for the present class of mesoporous solids, which are much more rigid than the metallic systems considered in Ref. 36, the scenario discussed by these latter authors is also applicable. In fact, assuming lattice strains to be caused by the pressure in the confined fluid alone, Prass et al. were able to extract realistic values of the nanomechanical properties of the mesoporous matrix [12]. This experimental observation sheds some doubt on whether or not the more sophisticated analysis put forth in Ref. 36

FIG. 2 . 1 :

 21 FIG. 2.1: (a) Characteristic X-ray diffraction patterns measured isothermally as functions of gas pressure from P = 0 to the bulk vapor pressure at coexistence P 0 . Part (a) shows the azimuthally averaged diffraction profiles as a function of P/P 0 and the magnitude of the scattering vector q = (4π/λ) sin θ where 2θ is the scattering angle and λ is the X-ray wavelength. The sharp drop in intensity at P/P 0 ≃ 0.35 indicates capillary condensation. The shift of the 11 peak in this region is clearly seen in part (b) of the figure. From Ref. 21 -reproduced by permission from the PCCP owner societies.

Fig. 2 .

 2 Fig. 2.1(b)]. From data such as the ones shown in Fig. 2.1 one may calculate a lattice strain ε ≡a (P ) -a 0 a 0 = q hk (0) -q hk (P ) q hk (P ) (2.1)

  empty model slit-pore as described in detail in Ref.21.

  parameter α is introduced in Eq. (3.5) to vary the strength of the fluidsubstrate interaction relative to that of the fluid-fluid interaction.

  (3.1), (3.5), and (3.6)]

  of ln Ξ with respect to T , µ, A, or s z[31]. Therefore, the value of m does not matter as one can verify from Eqs. (4.1)-(4.6) and as one would have anticipated for an equilibrium situation all along. The specific form of Eq. (4.5) results from an integration over momentum subspace formed by the N fluid molecules and 2N s substrate atoms each of which possess three translational degrees of freedom. Substrate atoms, which are bound to their individual equilibrium lattice sites [see Eq. (3.6)], are distinguishable such that only the factor 1/N! arises for the (indistinguishable) fluid molecules in the denominator of Eq. (4.5).

  fluid-particle coordinates [see, for example, Eq. (3.10) of Ref. 31].The integration over substrate-atom configurations reflects the thermal coupling between the substrate and the confined fluid which causes the deformability of the former. In other words, for finite κ the substrates cannot be treated as a static external field over the course of the simulation but are part of the system in the same spirit in which the specific configuration of a porous matrix enters the statistical physical analysis of quenched-annealed models of disordered mesoporous materials (see, for example, Chap. 7.2 of Ref. 2). The reader should also note that through the introduction of the harmonic binding potential in Eq. (3.6) we are effectively dealing with a macroscopically thick solid substrate where more remote lattice planes (beyond the one in immediate contact with the confined fluid phase) are taken into account in a continuum type of fashion through the stiffness parameter κ.In the actual Monte Carlo simulations we place N fluid molecules in a rectangular simulation cell of volume V = A 0 s z . For a typical (non-critical) liquid-like state N ≈ 700 depending on the choice of T and µ. In a standard simulation the walls comprise 2N s = 256 solid atoms such that each solid surface consists of n = 8 unit cells in both (x-and y-) directions corresponding to sidelengths s 0 = s x0 = s y0 ≃ 12.7 of the simulation box. We refer to a "Monte Carlo cycle" as a sequence of N + 2N s attempted sequential displacements of all particles followed by N attempted creations or destructions of fluid molecules where N denotes the actual number of molecules present in the system at the beginning of the cycle. Typical simulations comprise between 5 × 10 5 and 10 6 such cycles and are therefore rather long. The generation of a Markov chain in configuration space proceeds in a stan-

Figure 5 .of the 12 Page 12 of 27 URLFIG. 5 . 1 :

 5122751 Figure 5.1 confirms this expectation. For a subcritical isotherm T = 1.00 both experimental and theoretical data are plotted as functions of the reduced pressure P/P + 0 of the

"FIG. 5 . 2 :

 52 FIG. 5.2: Sorption strain ξ (•) (left ordinate) as a function of pressure P/P 0 in the bulk reservoir where the full line is a fit to the discrete data points. Also shown are data for P id zz ( ), P pot zz ( ), and P zz ( ) (right ordinate); (a) α = 1.0, (b) α = 1.5, (c) α = 2.0 [see Eq. (3.5)]. Page 15 of 27

5 . 8 )

 58 accounts for the departure of both solid atoms i and j from their respective equilibrium lattice sites. For reasons explained in Ref. 31 there is no contribution to P zz from solidsolid interactions. Because of the different functional forms of Eqs. (5.4) and (5.5) on one hand and (5.6) and (5.7) on the other hand, "force" and "virial" routes provide an internal consistency check on reliability of Monte Carlo simulations. Notice also that mechanical stability requires F[1] z

9 )FIG. 5 . 3 :

 953 FIG. 5.3: Local density ρ (z) as a function of position z between lower (z/s z0 = -0.5) and upper substrate (z/s z0 = +0.5); ( ) α = 1.0 µ = -12.15, ( ) α = 2.0, µ = -13.00 where κ = 30 and T = 1.0 in both cases.

FIG. 5 . 4 :

 54 FIG. 5.4: Pore strain ξ as a function of bulk pressure P on a logarithmic scale. Data are presented for various strengths of the fluid-substrate interaction α indicated in the figure [see Eq. 3.5)]. Solid line is a linear fit to guide the eye.

FIG. 5 . 5 :

 55 FIG. 5.5: As Fig. 5.4, but for various values of κ [see Eq. (3.6)]; (•) κ = 30, ( ) κ = 40, (△) κ = 50, ( ) κ = 100. Perfectly rigid substrates (κ = ∞) would be represented by a horizontal line ξ = 0.

20 Page 20 of 27 URL

 2027 3 of Ref. 2) G ≈ -As z0 P (5.11) Taking the exact differential of this expression (s z0 = const) and combining it with Eq. (5.10) the Gibbs-Duhem equation 0 = -SdT -Ndµ + As z0 dP + A 0 s z dP zz (5.12)obtains without much ado. Equation (5.12) expresses the fact that an equation of state of the form P zz = P zz T, µ, P exists in principle even though we do not know its functional form.

µFIG. 5 . 6 :

 56 FIG. 5.6: Plot of m versus 1/κ (•) [see Eq. (5.18)] using the Monte Carlo data presented in Fig. 5.5. Also shown is the dependence of m on α (•) using data presented in Fig. 5.3 where each point represents the slope of a best (linear) fit to an individual data set characterized by a specific value of α. Lines ( ) and ( ) are fits intended to guide the eye.

TABLE 5 .

 5 1: Comparison of virial and force expressions for P zz . Also listed is the associated pore strain ξ computed from Eq. (5.3).

	Eq. (5.6)	Eq. (5.4)

16 Page 16 of 27 URL

	Molecular Physics
	route leads to
	P zz = P id zz + P pot zz	(5.6)
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y

: http://mc.manuscriptcentral.com/tandf/tmph

  is required for mesoporous silica materials. In turn, the experimental results obtained in Ref. 12 validate the present numerical analysis.

	F o r F o r
	P P
	e e
	e r e r
	R R
	e e
	v i e v i e
	w w
	O n l O n l
	y y

Acknowledgments

We thank Oskar Paris (Montanuniversität Leoben) and Peter Fratzl (Max-Planck-Institut für Kolloid-und Grenzflächenforschung) for many fruitful discussions and for providing the experimental data presented in Fig. 2.1. Financial support from the Deutsche Forschungsgemeinschaft (Collaborative Research Area 448 "Mesoscopically structured composites") is gratefully acknowledged.