Throughout this article the term ‘function’ is defined as any biological function being assayed, i.e. behavior, reproduction, stress resistance, etc.

Michael S. Grotewiel, Virginia Commonwealth University, Department of Human Genetics, Richmond, VA 23298, USA. Tel.: +1 804 8289632 × 160; fax: +1 804 8283760; e-mail: msgrotewiel@vcu.edu

Abstract

Declines in function^{1} are common manifestations of aging in many phyla (Arking, 1998). As functional senescence is thought to drive the increasing risk of death with age, understanding functional senescence is important for understanding aging. Experimental investigation of functional senescence requires one to quantitate and compare age-dependent declines in function between cohorts. Such quantitation and comparison is often difficult owing to complexities in functional senescence data sets. Here, we discuss issues related to describing and contrasting age-related declines in function.

We parameterized functional senescence data in simple ways to generate descriptors for (1) the rate of functional decline, (2) the time to onset of functional decline and (3) total function. To illustrate how these descriptors can be used, we analyzed a hypothetical data set and one of our previously published data sets (Goddeeris et al., 2003). We conclude that no one descriptor sufficiently characterizes functional senescence. Useful distinctions between functional senescence in different cohorts can be made, however, when multiple descriptors are used in an integrated fashion.

Declines in function^{1} are common manifestations of aging in many phyla (Arking, 1998). As functional senescence is thought to drive the increasing risk of death with age, understanding functional senescence is important for understanding aging. Experimental investigation of functional senescence requires one to quantitate and compare age-dependent declines in function between cohorts. Such quantitation and comparison is often difficult owing to complexities in functional senescence data sets. Here, we discuss issues related to describing and contrasting age-related declines in function.

We parameterized functional senescence data in simple ways to generate descriptors for (1) the rate of functional decline, (2) the time to onset of functional decline and (3) total function. To illustrate how these descriptors can be used, we analyzed a hypothetical data set and one of our previously published data sets (Goddeeris et al., 2003). We conclude that no one descriptor sufficiently characterizes functional senescence. Useful distinctions between functional senescence in different cohorts can be made, however, when multiple descriptors are used in an integrated fashion.

Results and discussion

We present four data sets to illustrate some of the complexities in interpreting functional senescence data (Fig. 1A). The control data are derived from our previously published studies on senescence of negative geotaxis in wild-type flies (Goddeeris et al., 2003). The divergent, convergent and parallel data sets are hypothetical; they were designed to represent three major types of actual results that might be consistent with ameliorated functional senescence as compared with the control (Fig. 1A). The divergent group had the same peak function as control, but declined more slowly. The convergent group had a lower peak value than control, but declined so that its function at the last assessment was indistinguishable from control. The parallel group had elevated negative geotaxis scores at all ages so that its curve was parallel to control. Although statistical tests such as analysis of variance (anova) can be used to compare overall function across age in data sets like these (e.g Goddeeris et al., 2003), this approach leaves several questions unanswered, including: (1) Does the rate of functional decline differ between the data sets? (2) Is the time to onset of functional decline changed? (3) Is total function altered? We calculated several summary statistics (descriptors) from each of these hypothetical data sets toward addressing these questions.

First, we considered descriptors for the rate of functional decline. As a starting point, we calculated the absolute rate of decline (aROD, negative value of slope) from the hypothetical data (Fig. 1A). This descriptor depicts the absolute change in function per unit time determined from each assessment interval. Following the assumptions in our example data sets, the aRODs were lower overall for the divergent and convergent data sets relative to control, whereas this descriptor was identical in the parallel and control data (Fig. 1B). We also determined the proportional rate of decline (pROD) on the example data (Fig. 1A) using the formula ((F_{o} − F_{i})/F_{o}) × 100% where F_{o} is the functional value at the beginning of each interval and F_{i} is the value at the end of each interval. This descriptor represents the proportion of a function that is lost during each interval. As designed, pROD was constant in the divergent and convergent sets with the divergent set being lower overall (Fig. 1C). Additionally, pROD was lower in the parallel set than in control (Fig. 1C).

aROD and pROD naturally have different constraints based on their mathematical definitions. The maximum potential value of aROD is directly proportional to the magnitude of the function being investigated. Thus, large values for this descriptor might be artifactual in cohorts with peak function greater than control and small values might be artifactual in cohorts with peak function lower than control. From the perspective that a reduction in aROD could be interpreted as a reduction in the rate of functional senescence, this descriptor is conservative for groups with peak function equal to or greater than control and less conservative in groups with peak function lower than control. pROD, by contrast, varies inversely with the absolute magnitude of the function. Given groups with similar aRODs, pROD is higher in those with lower function (compare control with parallel and convergent with divergent, Fig. 1C). From the viewpoint that a reduction in pROD could be interpreted as a reduction in the rate of functional senescence, pROD is a conservative descriptor in groups with peak function similar to or lower than control but is less conservative in groups with peak function greater than control.

Second, we considered a descriptor for the time to onset of functional decline. Arking & Wells (1990) defined a loss-of-function constant (here called decline time, DT) as the time required for function to decline to 50% of its peak value. To explore the usefulness of this descriptor further, we determined the DT_{75} and DT_{50} (time required for function to decline to 75% and 50% of its peak value, respectively) on the data in Fig. 1(A). Values for DT_{75} and DT_{50} were interpolated from second-order polynomial curve fits (the least complicated curve that fits all the data). As expected, both DT_{75} and DT_{50} were increased for the divergent set relative to control (Fig. 1D). These two measures were also increased in the convergent and parallel sets, although not as robustly as in the divergent set (Fig. 1D). Groups with increased DT_{50} and DT_{75} as in the divergent, convergent and parallel sets would be good candidates for having extended periods during which function remains high relative to peak function for each cohort.

Finally, we considered a descriptor for total function throughout an experiment. As expected, total function (calculated as the area under the curve) was decreased (28%) in the convergent group and increased in the divergent (58%) and parallel (84%) groups relative to control. Groups such as the divergent and parallel sets could be interpreted to have increased total function, whereas groups such as the convergent set could be interpreted to have decreased total function.

When used together, our descriptors should provide a robust characterization of functional senescence data sets (Table 1). In contrast, comparing functional senescence between cohorts by using any single descriptor in isolation might be misleading because some descriptors might change while others do not and certain descriptors might exhibit contradictory changes. We would be confident that functional senescence has been slowed without obvious trade-offs in cohorts with reduced absolute and proportional rates of decline, extended decline times, and enhanced total function (e.g. divergent set). This is a straightforward example in which all of the descriptors have been enhanced. A more complicated case is the parallel set. In this case, pROD is reduced while aROD remains unchanged. If pROD exclusively were considered, one would conclude that the rate of functional senescence is slowed; if aROD exclusively were considered, the interpretation would be that the rate of functional senescence is unchanged. Neither of these interpretations is adequate because both ignore other information. By viewing all four descriptors together, one could reach a more satisfactory conclusion: although it remains ambiguous whether the rate of decline is altered, the parallel set exhibits extended decline times and enhanced total function that likely stem from an overall elevation in function. Thus, the parallel set would have meaningful positive changes in functional status within the context of aging. A final example is the convergent set. This example has decreased absolute and proportional rates of decline and extended decline times, but reduced total function. Such results would suggest that rate of functional senescence is slowed, but at the cost of reduced total functional capacity. This would indicate that an important trade-off has occurred in this group.

Table 1. Summary of analyses on theoretical data. Data in Fig. 1 were analyzed as described in the text. Downward arrows indicate reductions in the descriptor; upward arrows indicate increases. Major interpretations are listed for each of the groups

Group

aROD

pROD

DTs

Total function

Interpretations

Divergent

↓

↓

↑

↑

slows functional senescence and enhances total function

Convergent

↓

↓

↑

↓

slows functional senescence at the cost of total function

Parallel

unchanged

↓

↑

↑

Hyperfunctional at all assessments, positively impacts most descriptors of function across age

We recently reported that reduced expression of βPS, the mys gene product, ameliorates age-dependent senescence of negative geotaxis in Drosophila (Goddeeris et al., 2003). Here we report values for each of the descriptors calculated from these previously published data. aROD in mys^{xG}/+, mys^{nj}/+(Fig. 2C) and mys^{ts2} (data not shown) flies was reduced relative to control during the first two intervals and converged with control during later intervals. Similarly, pROD was initially reduced in mys^{xG}/+, mys^{nj}/+ (Fig. 2D) and mys^{ts2} animals (data not shown), but converged with control at later intervals. The consistent changes in aROD and pROD in all three mys hypomorphs suggest that the rate of functional decline is reduced by mutations in mys. Additionally, all three mys mutants had significantly increased DT_{75} and DT_{50} values (Fig. 2E and data not shown) as well as elevated total negative geotaxis (Fig. 2F and data not shown). It is interesting that the beneficial effects of mys mutations on the rates of functional senescence appeared during the first few weeks of adult life. Although this change in the rate of functional senescence occurred only for a relatively short time, the resulting favorable effects on negative geotaxis were evident for a considerable time thereafter. Importantly, the mys mutants have increased total negative geotaxis, confirming that reduced expression of βPS has positive consequences on total negative geotaxis function during the first eight weeks of life. Together, the reduced rates of functional decline, extended periods of high function and enhanced total function indicate that mys mutations confer large beneficial effects on senescence of negative geotaxis.

Our proposed descriptors provide a framework to characterize age-related declines in many functions (Arking & Wells, 1990; Le Bourg & Minois, 1999; Cook-Wiens & Grotewiel, 2002). It is possible that a multitude of treatments will change at least one of the descriptors in a positive way. We suggest that manipulations conferring the greatest beneficial effects can be identified when they decrease absolute and proportional rates of functional decline, extend decline times and enhance total function. Such determinations can be made only by evaluating multiple descriptors in an integrated fashion.

Acknowledgments

We thank Scott Pletcher and Marc Tatar for helpful discussions, Poonam Bhandari for comments on the manuscript, and Harvey Motulsky for help with statistical analyses. This work was supported by grants from NIH (MH64160, AG21199) and by the Neuroscience Program at Virginia Commonwealth University.