Goodness of fit to a mathematical model for Drosophila sleep behavior is reduced in hyposomnolent mutants

Characterization of wild type sleep

Sleep in wild type animals is consistent with that seen in the literature, in terms of both total time slept and sleep architecture (Stavropoulos & Young, 2011; Pfeiffenberger & Allada, 2012). Total sleep in control is 396.4 min in the day, with SD = 82.9, and 672.6 min in the night, with SD = 29.9 (Fig. 1A). Total daily sleep is 1069.0 min, with SD = 81.7. Mean sleep bout length in control is 35.1 min in the day, with SD = 14.4, and 216.1 min in the night, with SD = 118.2 (Fig. 1B). Mean sleep bout count in control is 12.6 in the day, with SD = 4.6, and 4.9 in the night, with SD = 4.2 (Fig. 1C).

Characterization of insomniac sleep

Insomniac demonstrates a robust phenotype in terms of total time slept. Insomniac animals tested in this experiment sleep significantly less than controls in the 24-hour period. Mean total sleep in insomniac is 770.1, with SD = 210.0, compared to 1069.0 with SD = 81.7 in wild type (Fig. 1A). According to a two-tailed, two-sample heteroscedastic (allowing for unequal variance) Student’s T-test, probability that measures of total sleep per 24 h in insomniac and controls came from the same distribution is given by p < 0.0001. Separate consideration of daytime and nighttime sleep reveals that nighttime total sleep in insomniac, at 394.7 (SD = 148.7) is significantly less than nighttime sleep in wildtype, at 672.6 (SD = 29.9) (Fig. 1A). Daytime total sleep in insomniac is unchanged as compared to control (Fig. 1A).

Insomniac also demonstrates a strong phenotype in sleep architecture. Bout length is shorter and bout count greater in insomniac as compared to its control. Mean sleep bout length in insomniac is 17.4 in the day, with SD = 8.4, and 30.0 in the night, with SD = 21.4 (Fig. 1B). Both of these values are significantly reduced as compared to wild type. Meanwhile, bout count is significantly greater in insomniac, with mean bout count = 23.1 (SD = 8.1) in the day and 16.8 (SD = 7.5) in the night (Fig. 1C).

That sleep in insomniac is poorly consolidated can be observed qualitatively. Figure 2 represents activity in insomniac and control. We see that, in the case of insomniac, activity is distributed throughout periods in which control flies normally sleep.

Production of a mathematical model

Figure 3 shows sleep bout count and sleep bout length for each animal-day or animal-night pair in either insomniac or control lines.

Sleep behavior is most regular in the case of control night (Fig. 3C). To this set of data, I fit the model (1)${\displaystyle y=a\cdot x^b}$ where y corresponds to mean sleep bout length, for an individual animal, over the course of a single night; and x corresponds to sleep bout count for that same individual animal over the course of a single night.

I use the coefficient of determination R² to assess the strength of this model. R², in short, describes the vertical distance from the data points to the line that attempts to approximate them. If the line provides a good approximate, the data points will tend to be close to the line, and so this vertical distance will tend to be small. A high R² indicates a low vertical distance, on the whole, and thus it indicates that the line of fit approximates the data well.

The coefficient of determination R² is 0.993 in the case of control night, indicating that Eq. (1) provides a good approximate to these data. In Fig. 3, Eq. (1) is fit to all experimental conditions. R² is not as high in other experimental conditions as it is in control night (Fig. 3C), indicating a worse fit to the model in these other experimental conditions.

Equation (1), the parameters that comprise it, and the R² coefficient might provide valuable insight towards the analysis of sleep behavior in Drosophila, even in experimental conditions where R² is relatively low.

The parameter b is negative in experimental conditions. This indicates that, as bout count rises, mean sleep bout length falls. Further, b tends to reside near −1.

In Eq. (1), a tends to estimate total sleep. For example, in Fig. 3C, a = 682.9. Consistent with this prediction, measured sleep for this genotype and timeframe is 672.6 min. Given the form of Eq. (1), one ought to expect that the parameter a would estimate total sleep. Suppose, in the regression Eq. (1), it so happens that b = − 1 exactly. Then we can re-express the equation as (2)${\displaystyle y\cdot x=a.}$ Equation (2) shows that (in the case b = − 1) the best regression generates a fixed constant a with the special property that the product of any pair of values attained by the variables x and y tends to fall close to a. These values in turn correspond to the bout count and mean bout lengths, respectively, of the animals. And, we know that, in an individual animal-time period pair, mean bout length times bout count equals total sleep for that time period. Thus we see why, when b falls close to −1, a estimates total sleep.

As b deviates from −1, a becomes a worse estimate of mean total sleep. For example, in Fig. 3B, a = 68. This drastically underestimates total sleep for this genotype and timeframe. For b > − 1, a is an underestimate of mean total sleep. For b < − 1, a is an overestimate. The tendency of a to estimate total sleep, as well as the relationship between b and a I have just described, holds in both control animals and in insomniac. In insomniac, a may not be as good an estimate of total sleep, in part because b may stray further from −1.

The coefficient of determination R² may be of use. As described earlier, R² is greatest in the setting of control sleep behavior at night. R² close to 1 indicates that the mathematical model closely fits the data.

R² is closer to 1 in the nighttime, as compared to the daytime, with genotype controlled for. In other words, control night has greater R² than control day; meanwhile, insomniac night has greater R² than insomniac day. Additionally, R² is farther from 1 in insomniac, as compared to control, with time of day controlled for. Insomniac night has lower R² than control night; insomniac day has lower R² than control day.

So, in the daytime, and in insomniac, the model tends to fit the data less well.

Under conditions where R² is relatively low, such as insomniac day, 95% confidence intervals for parameters a and b tend to be wider relative to the absolute value of these parameters. Also, 95% confidence bands tend to be wider as well in conditions with low R².

Statistical tests for the appropriateness of the mathematical model

Dependency between parameters a and b as they fit to the sleep data ranges from 0.822 to 0.984.

The sleep data do not pass the D’Agostino & Pearson omnibus K2 test of normalcy test in any genotype or timeframe, including control day, control night, insomniac day, and insomniac night.

Application of the model to activity data

I conducted a similar statistical analysis on the behavior of the animals used in this experiment, except considering activity bouts as opposed to sleep bouts.

Equation (1) does not fit the activity bout data as well as it fits the sleep bout data. R² is 0.608 at maximum.

Like in the case of the sleep bout data, R² is higher in control than it is in insomniac. R² values are 0.635 and 0.637 in control, daytime and nighttime, respectively (Figs. S1A and S1C), compared to 0.408 and 0.325 in insomniac (Figs. S1B and S1D).

Note that, in contrast with the sleep bout data, it is not the case in the activity bout data that R² changes in daytime as compared to nighttime. Within a given genotype, daytime and nighttime R² values are nearly identical.

Application of the model to sleep rebound

Control and insomniac animals were deprived of sleep throughout the entire 12-hour lights off period on the fourth night of an experiment. On the fifth day, both control and insomniac animals demonstrate sleep rebound, but insomniac sleep rebound is diminished as compared to wildtype (Fig. 4A). On night four of the sleep deprivation experiment, control slept 9.1 min on average (SD = 21.6), which is 486 min fewer than the mean sleep time for the first three nights of the experiment. Meanwhile, on night four, insomniac slept 3.4 min on average, which is 514 min fewer than the mean sleep time for the first three nights of the experiment. Then, on day five, following deprivation, control slept 496 min on average, with SD = 50.2. This is 100 min greater than the mean sleep time in control for the first three days of the experiment. insomniac, on day five, slept 496 min on average, with SD = 126. This is 70.6 min greater than the mean sleep time in insomniac for the first three days of the experiment.

I tested my model on the daytime sleep behavior during this fifth day, immediately following sleep deprivation. The results are shown in Figs. 4B and 4C. Equation (1) predicts sleep behavior well during sleep rebound. In fact, R² is higher following rebound than it is during usual daytime sleep for each respective genotype. Compare Fig. 4B with Figs. 3A and 4C with Fig. 3B.

The finding discussed in ‘Production of a mathematical model’, where R² is lower in insomniac than it is in control, still holds in the case of sleep rebound.

Application of the model to other lines

Trikinetics data from two other lines was obtained from Dr. William Joiner. These are iso31, a quasi-wildtype line which serves as control, and fumin, a hyposomnolent line. Two experiments, both conducted in parallel, compare sleep behavior in iso31 with sleep behavior in fumin.

Sleep time in control 500.1 min in the day with SD = 56.1, and 559.5 min at night with SD = 49.5 in the first experiment. In the second, sleep time in control is 508.4 min in the day with SD = 46.5, and 561.6 min at night with SD = 49.6. Sleep time is drastically reduced in fumin, more than it is in insomniac. In the first experiment, sleep is 199.4 min in the day (SD = 80.3) and 142.1 min at night (SD = 115.4). In the second, sleep is 198.7 min in the day (SD = 90.24) and 92.7 min in the night (SD = 99.1).

Regarding R², similar results to those described in ‘Production of a mathematical model’ are also found in Dr. Joiner’s data.

R² for iso31 is very high, with R² = 0.890 in the daytime and R² = 0.980 in the nighttime, in the first experiment (Figs. S2A and S2C). In the second experiment, R² is 0.921 and 0.960 in daytime and nighttime respectively. R² in fumin, on the other hand, is lower even than the R² values observed in insomniac. In the first experiment, R² = 0.009 and 0.058 in the daytime and nighttime, respectively (Figs. S2B and S2D). In the second, R² = 0.022 and 0.006 in the daytime and nighttime, respectively.

As in ‘Production of a mathematical model,’ R² tends to be greater for nighttime sleep than for daytime sleep. This is true in the first experiment universally, and in the second experiment in control but not in fumin.

Evaluation of sleep behavior

Sleep behavior in control is normal quantitatively (Fig. 1) and qualitatively (Fig. 2). This indicates that my sleep system is in good working order. Further, the sleep phenotype I have demonstrated in insomniac mutants, which is characterized by reduced total sleep and poor consolidation, is consistent with past reports (Stavropoulos & Young, 2011; Pfeiffenberger & Allada, 2012).

Merits of the mathematical model

Coefficient of determination R², which measures goodness of fit to the mathematical model described in Eq. (1), is as high as 0.993. This serves to validate the mathematical model: at least in some circumstances, the model describes behavior very well. Even in conditions where R² is not as high, such as in insomniac night or control day, the model appears to describe the behavior reasonably well considering the higher degree of variability within those data.

Note that R² constitutes a measure of sleep behavior independent of those measures usually studied in Drosophila sleep research, namely, total sleep, mean bout length, and mean bout count. Any of these measures could be changed in a Drosophila line, without change in R². Likewise, R² could theoretically change without corresponding change in total sleep, mean bout length, or mean bout count. Thus, the R² measure offers novelty.

As an alternative to R², the model also yields 95% confidence intervals for parameters a and b. As mentioned in the results, where R² is relatively low, the 95% confidence intervals for parameters a and b tend to be wide relative to the absolute value of these parameters. So, Eq. (1) parameter confidence interval width could also serve as a novel measure of sleep dysregulation. Confidence bands also tend to be wider in situations with low R².

Limitations of the model

Dependency between parameters a and b, when used to describe sleep behavior in insomniac and control animals, can be as high as 0.984. This indicates that a and b may be redundant. If a simpler model is desired, Eq. (2) would suffice. However, the inclusion of b seems to be merited, because production of a model conforming to Eq. (1) is not difficult, and b still improves goodness of fit.

That the sleep data universally fail the D’Agostino & Pearson omnibus K2 test under all circumstances might be cause for concern. Regardless of this finding, though, my model still appears to have merit, discussed in ‘Merits of the mathematical model’. Further, failure of this test need not indicate that nonlinear regression is an inappropriate strategy. Especially in large data sets, deviations from normalcy may reach statistical significance without corresponding to real practical meaning (D’Agostino, 1986). Therefore, it appears that my least-squares nonlinear regression procedure may be resistant to violations of the standard that underlying distributions be Gaussian (D’Agostino, 1986). Nevertheless, future work could look at the use of robust nonlinear regression models, as opposed to the least-squares nonlinear model used here. These are less distorted by data sets whose residuals come from non-Gaussian distributions (D’Agostino, 1986).

Also note that, if mean sleep bout length values are weighted by 1∕y², performance on the D’Agostino & Pearson omnibus K2 normalcy test is improved but still poor.

Teleological significance of the model

That Eq. (1) provides a good fit to the data—at least under some circumstances, like control night (Fig. 3C)—indicates that bout length and bout count are regulated with respect to each other such that, despite substantial variability in the values of these measures, total sleep tends to fall within a narrow range. Recall Eq. (2): if Eq. (2) provides a good fit to a set of animals, then no matter how much bout length and bout count vary among those animals, that each individual’s total sleep will be close to a.

So, R² may be indicative of how tightly bout length and bout count are regulated among a group of individuals so as to produce levels of total sleep within a narrow range. High R² could serve as a marker for successful regulation of sleep. A low R² would suggest decreased regulation of sleep or an interference with the ability to regulate sleep.

R² for daytime sleep is less than R² for nighttime sleep (‘Production of a mathematical model’). This might then suggest that daytime sleep is less tightly regulated than nighttime sleep.

R² for activity data is less than R² for sleep data (‘Application of the model to activity data’). This could suggest that time spent active is less tightly regulated than time spent asleep.

R² for sleep rebound is greater than R² for normal daytime sleep (‘Application of the model to sleep rebound’). This could suggest that sleep after a period of sleep deprivation is more strictly regulated than is sleep without sleep deprivation.

Finally, R² is low in insomniac (‘Production of a mathematical model’) and in fumin (‘Application of the model to other lines’), and it is lower in the latter than in the former. Meanwhile, mean total sleep is also lower in fumin than in insomniac. This suggests a sort of dose-dependent relationship between sleep impairment and R², where, as total 24-hour sleep falls, so does R². The more total sleep is impaired, the more achievement of tightly-regulated sleep also tends to be impaired.

Overall, these results suggest that R² could serve as a measure for the extent of sleep regulation. This is true in Drosophila as well as in higher animals.