The increased use of the COPAS Biosort in C. elegans
assays is driving a need to develop quantitative analysis techniques to make full use of these measurements. The Biosort can measure up to 100 nematodes/second 
, but the output measurements (EXT and TOF) have several characteristics that make it difficult to apply standard statistical analyses. First, extraneous materials such as clumps of bacteria, discarded cuticles, or chemical precipitates that are often aspirated along with C. elegans
are also measured by the Biosort. The measurements of extraneous material cannot always be distinguished from measurements of nematodes. This makes it difficult to estimate the numbers of aspirated nematodes at a given stage of growth. Second, although two measurements are made on each nematode (loading and aspiration) the measurements cannot be assigned to specific individual nematodes. This means growth rates for individual C. elegans
cannot be estimated. Finally, there is evidence suggesting that a portion of the nematode population curls while passing through the flow cell. Therefore, it is not possible to know when a particular TOF measurement corresponds to a curled or a straightened nematode.
The model presented in this paper deals with these issues by using a population model to estimate frequencies of nematodes in bins defined along log(EXT) or log(TOF) axes. Growth is modeled as a random variable with a probability distribution consisting of a discrete number of probabilities. In this way, the model is a discrete convolution, similar to a continuous convolution model recently used to describe cell growth 
. Using estimated frequency distributions, the model is able to estimate constant growth rates for log(EXT) and log(TOF) measurements, as well as the number of aspirated nematodes.
The population approach to modeling is also useful in dealing with the curling of the nematode. Ideally a nematode passes through the flow cell in a straightened position, so that TOF is highly correlated with the length of the animal. Curling disturbs this relationship producing a more variable distribution with more measurements for shorter TOFs. A quantitative analysis of TOF data that does not account for curling would underestimate nematode lengths and overestimate the natural variability in lengths.
A 48-h growth experiment showed additional low log(TOF) measurements for non-anesthetized nematodes, compared to anesthetized worms. These results are consistent with earlier reports of partial curling of a portion of nematodes affecting TOF measurements, as the relaxed, anesthetized nematodes are more successfully straightened by the faster moving core of sheath fluid in the flow cell 
. A ‘curling’ matrix with parameter values determined by the data was used to estimate the frequency distribution of nematode lengths. Applying the growth model to the estimated length distribution accurately fit the observed log(TOF) frequency distributions after optimization. Using the model with the log(TOF) values allowed the change points in growth to be based on the ‘uncurled’ TOF values.
Three-valued growth rates were estimated for populations with respect to both log(EXT) and log(TOF) measurements. Both growth rates showed an initial high rate of growth, possibly due to the addition of food to the loading population of starved L1 C. elegans
. During C. elegans
development, growth rates slowed twice at the estimated change points. Expected times to these change points using log(EXT) measurements occurred at 11 and 51 h, and at 7 and 34 h for log(TOF) measurements. Using the expected times of the estimated change points, biological changes that occurred at these times could be determined by direct microscopic examination. Under the conditions of this experiment (growth at 20°C in complete K-medium with sufficient food) C. elegans
at the earlier change points (7 and 11 h) were completing the L1/L2 molt. The 34 h TOF change point corresponded to the L3/L4 molt and C. elegans
were non-gravid young adults at the 51 h EXT change point (see Supporting Information S1
The results presented in this paper show that the model works well describing C. elegans growth from L1s to young adults. The growth rate over this interval is close enough to constant that the constraint on the predicted range of aspirated values either does not matter or accurately reflects observed behavior. To model growth from L4s to the point where adults stop growing, however, would require the model to have different structural parameters: either starting with a much larger number of smaller bins, or allowing nematodes to grow only 2, 3 or 4 bins, rather than 10, 11 or 12.
Using a Markov model, C. elegans growth can be quantitatively analyzed using medium- and high-throughout technologies. The goals of any mathematical model are two-fold: to provide a quantitative framework for analyzing the results of experiments and to increase insight into the biological nature of the observations. A framework for the first goal was provided by defining a mathematical structure whose predictions matched the biological observations. Using estimated parameters from the model, C. elegans growth rates could be quantified. In addition, it was shown that C. elegans growth rate varied during nematode development. The points where the growth rates changed corresponded to specific physiological events: theL3/L4 molt and the start of oogenesis in the young adult C. elegans. The flexibility and outputs of the C. elegans growth model make it ideal for investigating the response of C. elegans to environmental agents. In a companion paper, the model was used to characterize the effects of the organophosphate insecticide chlorpyrifos of C. elegans growth. It should be noted that the mathematical model developed in this paper can be applied to any population that is measured more than once and for which measurements on individual objects are not linked. Thus, this model could be applied to repeated flow cytometry measurements taken on cell populations.