Power of Linkage Disequilibrium Mapping to Detect a Quantitative Trait Locus (QTL) in Selected Samples of Unrelated Individuals

We considered a strategy to map quantitative trait loci (QTLs) using linkage disequilibrium (LD) when the QTL and marker locus were multiallelic. The strategy involved phenotyping a large number of unrelated individuals and genotyping only selected individuals from the two tails of the trait distribution. Power to detect trait‐marker association was assessed as a function of the number of QTL and marker alleles. Two patterns of LD were used to study their influence on power. When the frequency of the QTL allele with the largest effect and that of the marker allele linked in coupling were equal, power was maximum. In this case, increasing the number of QTL alleles reduced the power. The maximum difference in power between the two LD patterns studied was ～30%. For low QTL heritabilities (h²_QTL < 0.1) and single trait studies we recommend selecting around 5% of the upper and lower tails of the trait distribution.     

Design and Analysis of Genetic Association Studies to Finely Map a Locus Identified by Linkage Analysis: Assessment of the Extent to Which an Association Can Account for the Linkage

Association studies are often used to finely map quantitative trait loci identified by linkage analysis. Once a polymorphism associated with the trait has been identified, it may be useful to conduct linkage analyses which adjust for this polymorphism to determine the extent to which the association accounts for the linkage signal. However, methods for conducting statistical significance tests for an observed reduction in the linkage signal are not well developed. In the present study we develop methods for assessment of the statistical significance of an observed reduction in the variance due to the linked locus, with variance components or with Haseman-Elston linkage methods. Simulations indicate that these methods have appropriate levels of type I error and that, like other association statistics, their power depends on the magnitude of linkage disequilibrium between functional and marker alleles and on the extent of similarity between the frequency of the functional allele and the frequency of the associated marker allele. These methods can help determine which association results are likely due to strong linkage disequilibrium with functional alleles and, thus, can facilitate the selection of small chromosomal regions for more extensive study.     

Analysis of sports‐relevant polymorphisms in a large Brazilian cohort of top‐level athletes

In recent years, there have been an increasing number of genetic variants associated with athletic phenotypes. Here, we selected a set of sports‐relevant polymorphisms that have been previously suggested as genetic markers for human physical performance, and we examined their association with athletic status in a large cohort of Brazilians. We evaluated a sample of 1,622 individuals, in which 966 were nonathletes, and 656 were athletes: 328 endurance athletes and 328 power athletes. Only the AGT M268T minor allele was nominally associated with the endurance status. Conversely, we found that seven polymorphisms are more frequent in power athletes (MCT1 D490E, AGT M268T, PPARG P12A, PGC1A G482S, VEGFR2 Q472H, NOS3 C/T, and ACTN3 R577X). For all of these polymorphisms, power athletes were more likely than nonathletes or endurance athletes to carry the major allele or the homozygous genotype for the major allele. In particular, MCT1 D490E, AGT M268T, NOS3 C/T, and ACTN3 R577X showed stronger associations. Our findings support a role for these variants in the achievement of power athletic status in Brazilians: MCT1 D490E (T allele), AGT M268T (G allele), PPARG (C allele), PGC1A G482S (C allele), VEGFR2 Q472H (T allele), NOS3 C/T (T allele), and ACTN3 R577X (R allele).     

A Note on the Asymptotic Properties of Affected-sib-pair Linkage Tests

This article concerns the asymptotic properties of linkage tests for affected‐sib‐pair data under the null hypothesis of no linkage. We consider a popular single‐locus analysis model where the unknown parameters are the disease allele frequency, the three penetrances for the three genotypes at the disease locus, and the recombination fraction between the marker locus and the disease locus. These parameters are completely confounded under the null hypothesis of no linkage. We show that 1) If the total variance of the trait (i.e., the additive variance plus the dominance variance) is “separated” from 0, then the likelihood ratio statistic has an asymptotic 0.5χ²₀+ 0.5χ²₁ distribution; 2) If the prevalence of the trait is “separated” from 0 and the recombination fraction is fixed at 0, then the likelihood ratio statistic has an asymptotic distribution which is a mixture of χ²₀, χ²₁ and χ²₂ . The implications of these results are discussed.     

Combined Linkage and Association Mapping of Quantitative Trait Loci with Missing Completely at Random Genotype Data

In genetics study, the genotypes or phenotypes can be missing due to various reasons. In this paper, the impact of missing genotypes is investigated for high resolution combined linkage and association mapping of quantitative trait loci (QTL). We assume that the genotype data are missing completely at random (MCAR). Two regression models, “genotype effect model” and “additive effect model”, are proposed to model the association between the markers and the trait locus. If the marker genotype is not missing, the model is exactly the same as those of our previous study, i.e., the number of genotype or allele is used as weight to model the effect of the genotype or allele in single marker case. If the marker genotype is missing, the expected number of genotype or allele is used as weight to model the effect of the genotype or allele. By analytical formulae, we show that the “genotype effect model” can be used to model the additive and dominance effects simultaneously, and the “additive effect model” can only be used to model the additive effect. Based on the two models, F-test statistics are proposed to test association between the QTL and markers. The non-centrality parameter approximations of F-test statistics are derived to calculate power and to compare power, which show that the power of the F-tests is reduced due to the missingness. By simulation study, we show that the two models have reasonable type I error rates for a dataset of moderate sample size. However, the type I error rates can be very slightly inflated if all individuals with missing genotypes are removed from analysis. Hence, the proposed method can help to get correct type I error rates although it does not improve power. As a practical example, the method is applied to analyze the angiotensin-1 converting enzyme (ACE) data. Edited by Pak Sham.     

High Resolution T2 Association Tests of Complex Diseases Based on Family Data

This paper proposes family based Hotelling's T² tests for high resolution linkage disequilibrium (LD) mapping or association studies of complex diseases. Assume that genotype data of multiple markers or haplotype blocks are available for a sample of nuclear families, in which some offspring are affected. Paired Hotelling's T² test statistics are proposed for a high resolution association study using parents as controls for affected offspring, based on two coding methods: haplotype/allele coding and genotype coding. The paired Hotelling's T² tests take not only the correlation between the haplotype blocks or markers into account, but also take the correlation within each parent‐offspring pair into account. The method extends two sample Hotelling's T² test statistics for population case control association studies, which are not valid for family data due to correlation of genetic data among family members. The validity of the proposed method is justified by rigorous mathematical and statistical proof under the large sample theory. The non‐centrality parameter approximations of the test statistics are calculated for power and sample size calculations. From power comparison and type I error calculations, it is shown that the test statistic based on haplotype/allele coding is advantageous over the test statistic of genotype coding. Analysis using multiple markers may provide higher power than single marker analysis. If only one marker is utilized the power of the test statistic based on haplotype/allele coding is nearly identical to that of 1‐TDT. Moreover, a permutation procedure is provided for data analysis. The method is applied to data from a German asthma family study. The results based on the paired Hotelling's T² statistic tests confirm the previous findings. However, the paired Hotelling's T² tests produce much smaller P‐values than those of the previous study. The permutation tests produce similar results to those of the previous study; moreover, additional marker combinations are shown to be significant by permutation tests. The proposed paired Hotelling's T² statistic tests are potentially powerful in mapping complex diseases. A SAS Macro, Hotel_fam.sas, has been written to implement the method for data analysis.     

Adenosine is not a direct GHSR agonist--artificial cross-talk between GHSR and adenosine receptor pathways

Aim: To assess if adenosine is a direct growth hormone secretagogue receptor (GHSR) agonist by investigating the mechanism behind adenosine induced calcium release in human embryonic kidney 293s (HEK) cells expressing GHSR. Methods: Calcium mobilization, cyclic adenosine monophosphate (cAMP) and IP₃ experiments were performed using HEK cells stably expressing GHSR and/or adenosine A_2B receptor (A_2BR). Results: Adenosine has been widely reported as a GHSR agonist. In our hands, adenosine and forskolin stimulated calcium release from IP₃ controlled stores in HEK–GHSR cells but not in non‐transfected HEK cells. This release was not accompanied by increased IP₃ levels. The calcium release was both cholera toxin and U73122 sensitive, indicating the involvement of both Gα_s/adenylyl cyclase and Gα_q/11/phospholipase C pathways. Importantly, the GHSR inverse agonist [D‐Arg¹ D‐Phe⁵ D‐Trp^7,9 Leu¹¹]‐Substance P (SP‐analogue) blocked the adenosine stimulated calcium release, demonstrating that GHSR is involved. Assessment of the GHSR‐dependent calcium release using adenosine receptor agonists and antagonists resulted in a rank order of potencies resembling the profile of A_2BR. A_2BR over‐expression in HEK–GHSR cells enhanced potency and efficacy of the adenosine induced calcium release without increasing IP₃ production. Moreover, A_2BR over‐expression in HEK cells potentiated NECA‐induced cAMP production. However, GHSR expression had no effect on intracellular cAMP production. Conclusion: In HEK–GHSR cells adenosine activates endogenously expressed A_2BR resulting in calcium mobilization. We hypothesize that the responsible mechanism is cAMP‐dependent sensitization of IP₃ receptors for the high basal level of IP₃ caused by GHSR constitutive activity. Altogether, our results demonstrate that adenosine is not a direct GHSR agonist.     

EMK: A Novel Program for Family-Based Allelic and Genotypic Association Tests on Quantitative Traits

The QTDT program is a widely‐used program for analyzing quantitative trait data, but the methods mainly test allelic association. Since the genotype of a marker is a direct observation for an individual, it is of interest to assess association at the genotypic level. In this study, we extended the allele‐based association method developed by Monks and Kaplan (MK method) to genotype‐based association tests for quantitative traits. We implemented a novel extended MK (EMK) program that can perform both allele‐ and genotype‐ based association tests in any pedigree structure. To evaluate the performance of EMK, we utilized simulated pedigree data and real data from our previous report of GSTO1 and GSTO2 genes in Alzheimer disease (AD). Both allele‐ and genotype‐based EMK methods (allele‐EMK and geno‐EMK) showed correct type I error for various pedigree structures and admixture populations. The geno‐EMK method showed comparable power to the allele‐EMK test. By treating age‐at‐onset (AAO) as a quantitative trait, the EMK program was able to detect significant associations for rs4925 in GSTO1 (P= 0.006 for allele‐EMK and P= 0.009 for geno‐EMK), and rs2297235 in GSTO2 (P= 0.005 for allele‐EMK and P= 0.009 for geno‐EMK), which are consistent with our previous findings.     

Beta2-adrenergic receptor allele frequencies in the Quechua, a high altitude native population

The beta₂‐adrenergic receptor is involved in the control of numerous physiological processes and, as the primary catecholamine receptor in the lungs, is of particular importance in the regulation of pulmonary function. There are several polymorphic loci in the beta₂‐adrenergic receptor gene that have alleles that alter receptor function, including two (A/G⁴⁶, G/C⁷⁹) that increase agonist sensitivity. As such a phenotype may increase vaso and bronchial dilation, thereby facilitating air and blood flow through the lungs, we hypothesized that selection may have favoured these alleles in high altitude populations as part of an adaptive strategy to deal with the hypoxic conditions characteristic of such environments. We tested this hypothesis by determining the allele frequencies for these two polymorphisms, as well one additional missense mutation (C/T⁴⁹¹) and two silent mutations (G/A²⁵² and C/A⁵²³) in 63 Quechua speaking natives from communities located between 3200 and 4200 m on the Peruvian altiplano. These frequencies were compared with those of two lowland populations, one native American (Na‐Dene from the west coast of Canada) and one Caucasian of Western European descent. The Quechua manifest many of the pulmonary characteristics of high altitude populations and differences in allele frequencies between the Quechua and lowlanders could be indicative of a selective advantage conferred by certain genotypes in high altitude environments. Allele frequencies varied between populations at some loci and patterns of linkage disequilibrium differed between the old‐world and new‐world samples; however, as these populations are not closely related, significant variation would be expected due to stochastic effects alone. Neither of the alleles associated with increased receptor sensitivity (A⁴⁶, G⁷⁹) was significantly over‐represented in the Quechua compared with either lowland group. The Quechua were monomorphic for the C allele at base 79. This variant has been associated with body mass index; however no clearly defined metabolic phenotype has been established. In addition, we sequenced the coding region of the gene in three unrelated Quechua to determine if there were any other polymorphisms common in this population. None were detected.     

Natural Genetic Variation Underlying Differences in Peromyscus Repetitive and Social/Aggressive Behaviors

Peromyscus maniculatus (BW) and P. polionotus (PO) are interfertile North American species that differ in many characteristics. For example, PO exhibit monogamy and BW animals are susceptible to repetitive behaviors and thus a model for neurobehavioral disorders such as Autism. We analyzed these two stocks as well as their hybrids, a BW Y^PO consomic line (previously shown to alter glucose homeostasis) and a natural P. maniculatus agouti variant (A^Nb = wide band agouti). We show that PO animals engage in far less repetitive behavior than BW animals, that this trait is dominant, and that trait distribution in both species is bi-modal. The A^Nb allele also reduces such behaviors, particularly in females. PO, F1, and A^Nb animals all dig significantly more than BW. Increased self-grooming is also a PO dominant trait, and there is a bimodal trait distribution in all groups except BW. The inter-stock differences in self-grooming are greater between males, and the consomic data suggest the Y chromosome plays a role. The monogamous PO animals engage in more social behavior than BW; hybrid animals exhibit intermediate levels. Surprisingly, A^Nb animals are also more social than BW animals, although A^Nb interactions led to aggressive interactions at higher levels than any other group. PO animals exhibited the lowest incidence of aggressive behaviors, while the hybrids exhibited BW levels. Thus this group exhibits natural, genetically tractable variation in several biomedically relevant traits.     

Further Investigation of Linkage Disequilibrium SNPs and their Ability to Identify Associated Susceptibility Loci

There is currently considerable interest in the use of single‐nucleotide polymorphisms (SNPs) to map disease susceptibility genes. The success of this method will depend on a number of factors including the strength of linkage disequilibrium (LD) between marker and disease loci. We used a data set of SNP genotypings in the region of the APOE disease susceptibility locus to investigate the likely usefulness of SNPs in case‐control studies. Using the estimated haplotype structure surrounding and including the APOE locus, and assuming a codominant disease model, we treated each SNP in turn as if it were a disease susceptibility locus and obtained, for each disease locus and markers, the expected likelihood ratio test (LRT) to assess disease association.We were particularly interested in the power to detect association with the susceptibility polymorphism itself, the power of nearby markers to detect association, and the ability to distinguish between the susceptibility polymorphism and marker loci also showing association. We found that the expected LRT depended critically on disease allele frequencies. For disease loci with a reasonably common allele we were usually able to detect association. However, for only a subset of markers in the close neighbourhood of the disease locus was association detectable. In these cases we were usually, but not always, able to distinguish the disease locus from nearby associated marker loci. For some disease loci, no other loci demonstrated detectable association with the disease phenotype. We conclude that one may need to use very dense SNP maps in order to avoid overlooking polymorphisms affecting susceptibility to a common phenotype.     

The 5‐HT2A receptor gene 102T/C polymorphism is associated with suicidal behavior in depressed patients

Several lines of evidence suggest that genetic factors constitute an important determinant of suicidal behavior. A significant association between the 5‐HT_2A‐C allele and suicidality has recently been reported. The aim of this study was to investigate whether the proposed association between 5‐HT_2A‐102T/C polymorphism and suicidality could be replicated in a larger and independent sample of Spanish patients with major depression. The 102T/C polymorphism of the 5‐HT_2A receptor gene was analyzed in 159 patients with major depression (DSM‐IV criteria) and 164 unrelated and healthy controls using a case control design. All individuals were subjects of Spanish origin. Significant differences in allele (chi‐square = 4.13, df = 1, P = 0.04) and genotype (chi‐square = 6.19, df = 2, P = 0.04) distributions were found between non–suicide attempters and suicide attempters. Moreover, those patients carrying 5‐HT_2A‐C allele had more than five times the risk for attempting suicide than noncarriers (OR = 5.50, 95% CI = 1.18–35.20, P = 0.01). Our results replicate the proposed association between 5HT_2A‐C allele and suicidality in major depression. Moreover, no overall associations are detected when patients with major depression and controls are compared for 102T/C frequencies, suggesting that the increased risk for suicidality conferred by 5‐HT_2A‐C allele is primarily associated with suicidal behavior and not with the diagnosis of major depression itself. © 2001 Wiley‐Liss, Inc.     

No evidence for an allelic association between schizophrenia and markers D22S278 and D22S283

We report a case control association study using markers D22S278 and D22S283 in 90 unrelated patients with DSMIII-R schizophrenia and 90 controls matched for ethnicity, age and sex. No differences between allele frequencies for either marker were observed when the two groups were compared (D22S278: χ² = 6.53, df = 7, P = 0.51; D22S283: χ² = 14.73, df = 15, P = 0.48). These findings fail to support previous work by others suggesting the presence of allelic association between the markers D22S278 and D22S283 and schizophrenia. Am. J. Med. Genet. 74:37–39, 1997. © 1997 Wiley-Liss, Inc.     

Localized rest and stress human cardiac creatine kinase reaction kinetics at 3 T

Changes in the kinetics of the creatine kinase (CK) shuttle are sensitive markers of cardiac energetics but are typically measured at rest and in the prone position. This study aims to measure CK kinetics during pharmacological stress at 3 T, with measurement in the supine position. A shorter “stressed saturation transfer” (StreST) extension to the triple repetition time saturation transfer (TRiST) method is proposed. We assess scanning in a supine position and validate the MR measurement against biopsy assay of CK activity. We report normal ranges of stress CK forward rate (k_f^CK) for healthy volunteers and obese patients. TRiST measures k_f^CK in 40 min at 3 T. StreST extends the previously developed TRiST to also make a further k_f^CK measurement during <20 min of dobutamine stress. We test our TRiST implementation in skeletal muscle and myocardium in both prone and supine positions. We evaluate StreST in the myocardium of six healthy volunteers and 34 obese subjects. We validated MR‐measured k_f^CK against biopsy assays of CK activity. TRiST k_f^CK values matched literature values in skeletal muscle (k_f^CK = 0.25 ± 0.03 s^?1 vs 0.27 ± 0.03 s^?1) and myocardium when measured in the prone position (0.32 ± 0.15 s^?1), but a significant difference was found for TRiST k_f^CK measured supine (0.24 ± 0.12 s^?1). This difference was because of different respiratory‐ and cardiac‐motion‐induced B₀ changes in the two positions. Using supine TRiST, cardiac k_f^CK values for normal‐weight subjects were 0.15 ± 0.09 s^?1 at rest and 0.17 ± 0.15 s^?1 during stress. For obese subjects, k_f^CK was 0.16 ± 0.07 s^?1 at rest and 0.17 ± 0.10 s^?1 during stress. Rest myocardial k_f^CK and CK activity from LV biopsies of the same subjects correlated (R = 0.43, p = 0.03). We present an independent implementation of TRiST on the Siemens platform using a commercially available coil. Our extended StreST protocol enables cardiac k_f^CK to be measured during dobutamine‐induced stress in the supine position.     

Tolylene Diisocyanate‐Based Poly(imido‐amide)s, 1. Microstructure and Distributions of Amide and Imide Groups studied by 1H and 13C NMR

Thermostable tolylene diisocyanate (D)‐based poly(imide‐amide)s (PIA) are obtained by homogeneous polycondensation at completion between D (a mixture of two isomers) and trimellic anhydride (T) in N,N′‐dimethylene urea solvent at 185°C. The study of the microstructures of these PIAs by ¹H and quantitative ¹³C NMR is efficiently aided by using (D + phthalic anhydride) and (D + benzoic acid) model condensates. In all cases, imide (I) groups are found to be slightly in deficit and traces of in‐chain amic acid groups (precursors of I) are present. With tolylene 2,4‐diisocyanate (D′), A (amide) and I groups are preferably formed at the ortho (or 2) and at the para (or 4) positions relative to the methyl group of the tolylene cycle respectively. The concentration of A²I⁴ sequences are found to equal two times that of I²A⁴. For tolylene 2,6‐diisocyanate (D″), the concentration of A²I⁶ mixed sequence amounts to three times that of A²A⁶ or I²I⁶. The variations of the concentrations of AⁱA^j, AⁱI^j, IⁱI^j sequences versus the composition of D (= x D′ + y D′′, x + y = 1) are perfectly linear. I being bulkier than A, the formations of I and A groups on tolylene cycles are found to be mainly governed by steric hindrance effects.     

The Ca2+-sensitive K+-currents underlying the slow afterhyperpolarization of bullfrog sympathetic neurones

Ca²⁺-sensitive K⁺ currents involved in the slow afterhyperpolarization (a.h.p.) of an action potential of bullfrog sympathetic neurones were studied with a single-electrode voltage clamp method. The outward tail current (I_AH) generated after the end of a depolarizing command pulse (from the holding potential of –60 mV to 0 mV, 5–20 ms in duration), mimicking an action potential, was separated into at least two exponential components (I_AHf and I_AHs). They were identified as K⁺ currents, since their reversal potentials were close to the K⁺ equilibrium potential and they were sensitive to external K⁺. The time constant of I_AHf (t _f; 44 ms at –60 mV) was decreased by membrane hyperpolarization from –40 to –80 mV, while that of I_AHs (t _s; 213 ms) remained constant. Removal of external Ca²⁺ or addition of Cd²⁺ significantly decreased the I_AHs amplitude (A_s) andt _f without a change int _s and the I_AHf amplitude (A_f). On the other hand, increasing Ca²⁺ influx by applying repetitive command pulses enhanced both A_f and A_s with negligible effects ont _f andt _s, and produced a much slower component. Intracellular injection of EGTA reduced A_f with no effect ont _f, and increased A_s with a decreasedt _s. Both muscarine and (±)-tubocurarine, which reduced I_AHs, hardly affected I_AHf. These results indicate that a.h.p. is induced by the activation of two distinct Ca²⁺-dependent K⁺ channels, which differ in voltage sensitivity, Ca²⁺-dependence and pharmacology.     

Combined high resolution linkage and association mapping of quantitative trait loci

In this paper, we investigate variance component models of both linkage analysis and high resolution linkage disequilibrium (LD) mapping for quantitative trait loci (QTL). The models are based on both family pedigree and population data. We consider likelihoods which utilize flanking marker information, and carry out an analysis of model building and parameter estimations. The likelihoods jointly include recombination fractions, LD coefficients, the average allele substitution effect and allele dominant effect as parameters. Hence, the model simultaneously takes care of the linkage, LD or association and the effects of the putative trait locus. The models clearly demonstrate that linkage analysis and LD mapping are complementary, not exclusive, methods for QTL mapping. By power calculations and comparisons, we show the advantages of the proposed method: (1) population data can provide information for LD mapping, and family pedigree data can provide information for both linkage analysis and LD mapping; (2) using family pedigree data and a sparse marker map, one may investigate the prior suggestive linkage between trait locus and markers to obtain low resolution of the trait loci, because linkage analysis can locate a broad candidate region; (3) with the prior knowledge of suggestive linkage from linkage analysis, both population and family pedigree data can be used simultaneously in high resolution LD mapping based on a dense marker map, since LD mapping can increase the resolution for candidate regions; (4) models of high resolution LD mappings using two flanking markers have higher power than that of models of using only one marker in the analysis; (5) excluding the dominant variance from the analysis when it does exist would lose power; (6) by performing linkage interval mappings, one may get higher power than by using only one marker in the analysis.     

The resonant step frequency in human running

 At running speeds less than about 13 km h^–1 the freely chosen step frequency (f _free) is lower than the frequency at which the mechanical power is minimized (f _min). This dissociation between f _free and f _min was investigated by measuring mechanical power, metabolic energy expenditure and apparent natural frequency of the body’s bouncing system (f _sist) during running at three given speeds with different step frequencies. The f _free requires a mechanical power greater than that at f _min mainly due to a larger vertical oscillation of the body at each step. Energy expenditure is minimal and the mechanical efficiency is maximal at f _free. At a given speed, an increase in step frequency above f _free results in an increase in energy expenditure despite a decrease in mechanical power. On the other hand, a decrease in step frequency below f _free results in a larger increase in energy expenditure associated with an increase in mechanical power. When the step frequency is forced to values above or below f _free, f _sist is forced to change similarly by adjusting the stiffness of the bouncing system. However the best match between f _sist and step frequency takes place only in proximity of f _free (2.6–2.8 Hz). It is concluded that during running at speeds less than 13 km h^–1 energy is saved by tuning step frequency to f _sist, even if this requires a mechanical power larger than necessary. Received: 13 January 1997 / Received after revision: 7 April 1997 / Accepted: 28 May 1997     

Free rotation dimensions of linear polymers and application to polyalkylene oxides and polyalkylene sulphides

A general expression has been derived for the mean square end-to-end distance 〈r²〉_0f of linear polymers in the unperturbed freely rotating state. For a chain having q bonds per segment, a degree of polymerisation p and a total of N (= pq) bonds in the chain, the expression may be written 〈r²〉_0f = N[A₀/q + AB/pq]. Here A₀ is the sum of scalar products of the bond vectors within a segment and A is the corresponding sum for pairs of bond vectors in adjacent segments. B denotes a series having a value dependent on p and the bond angles. The formula has been applied to the particular cases of [(CH₂)_m?X?]_p where X = oxygen, X = sulphur and m = 1, 2, 3, 4, 6, and 10. Calculations of [〈r²〉_0f/N]^1/2 have been made for values of p in the range 1 ? p ? ∞, and the variation of [〈r²〉_0f/N]^1/2 with both chain length and chain structure discussed.