Trajectories of Senescence through Markov Models
The theory of ageing currently has to get by without a useable operational definition of its central object, ageing. Instead, coarse proxies stand in for ageing, whether mortality or changes in fertility. Experiments designed by teams in California and Mexico are under way to study the life course of ageing in fruit flies, one of the standard model organisms. Flies in these experiments will have their behaviour continuously recorded over their entire lifespans, with the goal of understanding how changes in behaviour might signal underlying changes in the inherent senescence state.
This project will develop statistical methods to 1) organise and structure the data, 2) recognise the signal of senescence in this mass of data, and 3) link the senescence signal to mathematical models which have been central to the theoretical study of ageing. The basic object is a kind of Hidden Markov Model, adapted to this particular structure of observations. We will develop both the basic theoretical statistics of this model and a practical computational algorithm, test the procedure on simulated data, and help the experimental teams with the data analysis as the results become available.
David Steinsaltz, University of Oxford
Martin Kolb, University of Oxford
Partners and Collaborators:
Operational definition of ageing have proved elusive. It is presumed that there is a senescence state, correlated with but also distinct from calendar age, which drives the observable phenomena of ageing.
New experimental techniques are making possible a much higher level of resolution in studying physiological and behavioural changes in standard model systems for ageing. These experiments are already yielding masses of data for which there are currently no statistical tools that are designed to tease the signal of progressive ageing from the momentary random fluctuations.
Markov models are frequently applied in theoretical studies of ageing. There has been little opportunity to validate the applicability of a Markov model of ageing based on real data. Such validation - or demonstration of the inadequacy of such models - would have important ramifications on the theoretical side.
Statistical methods are also lacking for effectively understanding the link between ageing, population growth, and environment for wild populations subject to environmental fluctuations. Some of the same statistical tools being developed here may be applicable to studying the evolution of ageing in natural populations.
The overall aim of this project is to improve statistical methodology for analysing longitudinal ageing experiments with simple model organisms.
Validated model and fitting method: We will prove appropriate theorems and carry out simulation studies which should show that Gaussian approximation should provide appropriate estimates of the parameters in these models. Should this planned approach turn out to be inadequate, we will work on developing new estimation methods, perhaps using new innovations in quasi-likelihood approaches.
Analysis of ongoing fruitfly experiments: Data are slowly coming in from the experiments in Mexico. We will use these to refine our models, particularly as regards the best way of including daily cycles and the most appropriate model of mortality. We will fit the data to our model, hopefully yielding validated estimates of the senescence process in these flies. These results will hopefully feed back into the design of later rounds of experiments, and also provide a jumping-off point for analysis of the variability and predictability of senescence in C. capitata and related species.
Other applications of Markov switching model methodology: There are other data sets and other problems connected with ageing, to which related methods could be applied. We will develop mathematical tools for analysing the evolution of ageing in populations subject to randomly fluctuating environments, and statistical tools for analysing age- and stage-structured data from wild populations. We also hope to analyse newly available data on progressive calcification in rats with kidney failure.
Electronic Behavior Monitoring System: The apparatus has been designed and built at the Instituto Nacional de Astrofísica, Optica y Electrónica (INAOE) in Puebla, Mexico. Each of the three EBMS is capable of monitoring nine individual flies arranged in a 3 x 3 cage configuration (i.e. 27 flies in total). The operational steps include the following: (1) A single newly-eclosed fly is placed in each of the 9 cages within a system; (2) the digitised images of these individual flies are captured on a micro-second scale over a predetermined interval (e.g. 10, 20; 30 seconds; 1, 2, 3 mins) and stored; (3) the system records, on a scale of micro-seconds, the activity and behaviour of individual fruit flies throughout their adult lives including movement in space; (4) position records are combined with multiple digital pictures to classify the fly's behaviour (resting, moving in place, walking, flying, feeding, and drinking) and its location within the cage.
Markov switching model: Our plan is to adapt Markov switching models to these longitudinal data. We assume the existence of a hidden diffusion process specifying an implicit physiological "age" or senescent condition for the organism. The random development of this hidden variable drives the levels of markovian switching rates which specify probabilities of transitions between behavioural states, the transitions that comprise the data. The hidden variable serves as the predictor variable in a log-odds model for the switching rates.
Gaussian approximation and Kalman filter: One of the key problems in fitting these models is the difficulty of computing likelihoods efficiently enough to be able to optimise them. We will use a Gaussian approximation to the Markov likelihood, which will enable us to apply efficient Kalman filter techniques to compute approximate likelihoods.
Semiparametric mortality modelling: One key open question concerns the functional form linking senescence state with mortality rate. We will develop techniques to model senescence as a proportional hazards factor, allowing us to dispense with preselected functional forms.
1. Key policy and/or practice implications of the research
The project is expected to yield improved statistical approaches to longitudinal studies of ageing in model organisms.
2. Key non-academic user groups that will be targeted