The corresponding genotype frequencies at age x for the three genotypes in males are p(x)=(p2s(x)r2rgs=s(x)); p(x)=(m20;m1)(2p(1-p)s(x)r1rgs=s(x)); p(x)=(1-p)2s(x)/s(x)). In the same way, we can calculate p(x), p(x), and p(x) for females. Based on a multinomial distribution (Evans et al., 1993), the likelihood function can be written as the distribution of subjects by age group, sex, and genotype.

Table 2 presents estimated allele frequency and relative risks by genotype for each allele. Note that in Table 2, we only estimate one risk for carriers of allele 10 (non-significant) because we only observe one homozygous carrier of the allele in the sample (Table 1).

Fig. 1 plots the observed and estimated frequencies of alleles 8 and 10* carriers by age. While frequency for allele eight carriers is almost constant (Fig. 1A), the frequency of heterozygous genotypes for allele 10* is approximately the square root of the risk of the 10*10* homozygous genotype, thus suggesting a dose-effect for this allele. According to the model specification, allele 10* phenotypic value in term of log hazard of death is additive on the estimated relative risk. The frequency for 10* allele carriers climbs up as age advances (Fig. 1B), revealing obvious survival advantages for 10* allele carriers.

As for the other alleles, allele 9 seems to be recessive and detrimental because the homozygotes for this allele tend to have increased hazard of death over the ages (r=1.409 with SE=0.178, P=0.022). However, such a risk value becomes non-significant after adjustment for multiple comparisons.

Lastly, as for gene-sex interaction, we see from Table 2 that alleles 6 and 8 seem to decrease and allele 10* to increase the risks of death in males. However, also in this case, the P values become non-significant after adjustment for multiple comparisons.

Given the complexity of the longevity trait, a demanding issue is to develop new models that are capable of capturing the various gene trajectories relating to the trait which occur as the population ages and survival selection operates. The present study could serve as a hint in this direction because it introduces an analytical method that permits revealing the interaction between the alleles of a gene (TH) associated with human longevity. The method is applied to a large dataset so that subtle effects of the gene variant on the complex trait could be revealed.

The basic statistical model described in this paper has been introduced in recent publications (Yashin et al., 1998, 1999b, 2000; Tan et al., 2001b,c). However, the parameterization in the present application shows the feasibility of detecting an allele-allele interaction (see Table 2 and Fig. 1) which implies dominant and additive beneficial effects on the risk of death (allele 10*). An important feature of this application is the combination of individual genetic data with population demographic information in determining the relative risks of the observed genes. The baseline hazard function in this approach is estimated non-parametrically so that no specific survival distribution is imposed on the likelihood function (Tan et al., 200Al. (2001b). However, it can be seen from this study that the proportional hazard assumption is implicated in human longevity, and in an allele-specific way. How could these effects be explained from a biological point of view? TH encodes the rate-limiting enzyme for the synthesis of catecholamines, amino acid-derived molecules that act both as hormones (adrenaline) and neurotransmitters (dopamine and noradrenaline). The TH gene is therefore a master gene in the neuroendocrine system (NES). Vertebrates have developed a complex hierarchy of multi-organ NES, integrated with the immune system to form the immune-NES. This system governs the capability to cope with a variety of stress factors and, consequently, rate and quality of aging (Franceschi et al., 2000). The HUMTHO1 microsatellite acts, in vitro, as a transcription regulatory element showing a silencing effect which correlates with the number of repeats (Meloni et al., 2001). In particular, the inhibition is proportional to the number of repeats up to allele 8, while by increasing the number of repeats up to 10, no further inhibitory effect is observed but rather a relative enhancement depending on the cell lines used (Albanese et al., 2001). These findings support the idea that HUMTHO1 microsatellite regulates TH transcription in an allelic-specific way and modulates the activity of the TH gene acting as a quantitative genomic effector. These allele-specific effects would explain the dominant and dose-dependent beneficial effects we observed for allele 10 by our approach. This effect is not found for allele 10*, likely because of its low frequency with respect to the sample size. On the other hand, a large amount of data suggests the importance of an efficient production of catecholamines in healthy aging and longevity. Indeed, there are now numerous studies demonstrating that sympathetic factors play a central role not only in cardiovascular homeostatic control but also in the pathogenesis and/or in the progression of several cardiovascular diseases, such as essential hypertension (Rahn et al., 1999) and myocardial infarction (Bertel et al., 1982). Therefore, good efficiency in TH gene transcription may be a key for an appropriate response towards stochastic and age-related stress factors, and consequently for longevity. 

References 
Albanese, V., Faucon Biguet, N., Klefer, H., Bayard, E., Mallet, J., Meloni, R., Polymeropoulos, M.H., Xiao, H., Rath, D.S., Merrill, C.R. (2001). Quantitative effects on gene silencing by allelic variation at a tetranucleotide microsatellite. Hum. Mol. Genet., 10(17), 1785-1792.