Parameter discovery in stochastic biological models using simulated annealing and statistical model checking

Abstract

Stochastic models are increasingly used to study the behaviour of biochemical systems. While the structure of such models is often readily available from first principles, unknown quantitative features of the model are incorporated into the model as parameters. Algorithmic discovery of parameter values from experimentally observed facts remains a challenge for the computational systems biology community. We present a new parameter discovery algorithm that uses simulated annealing, sequential hypothesis testing, and statistical model checking to learn the parameters in a stochastic model. We apply our technique to a model of glucose and insulin metabolism used for in-silico validation of artificial pancreata and demonstrate its effectiveness by developing parallel CUDA-based implementation for parameter synthesis in this model.

Keywords: CPS; CUDA; SPRT; artificial pancreata; behavioural specifications; biochemical systems; bioinformatics; biomedical devices; computational systems biology; cyber–physical systems; glucose–insulin model; machine learning; parameter discovery; parameter synthesis; probabilistic verification; simulated annealing; statistical hypothesis testing; statistical model checking; stochastic modelling; temporal logic.

Figure 1

Problem definition. Given the behavioural specifications about the system being modelled, the parameterised stochastic models, the parameter space and a correctness confidence, our algorithm seeks to discover the value of the parameters that enables the given model to satisfy the given probabilistic behavioural specifications

Figure 2

Stochastic models. Our algorithm is applicable to both discrete and continuous time stochastic models. CTMCs are particularly important for studying biochemical systems, while ODEs interacting with random variables naturally model cyber-physical biomedical systems

Figure 3

Our SPRT-based parameter discovery procedure. Given a probabilistic behavioural specification and a parameterised stochastic model, the SPRT procedure decides if the model satisfies the specification

Figure 4

Classic simulated annealing algorithm. The figure shows the classic algorithm that uses the exact probability calculation for simulated annealing

Figure 5

SPRT-based simulated annealing algorithm for parameter discovery. Our new algorithm for parameter discovery that combines the SPRT with simulated annealing

Figure 6

Results of the parameter synthesis algorithm (I). The figure shows the first parameter (namely pancreatic responsivity to glucose rate of change) of the glucose-insulin model whose value was discovered by our algorithm (see online version for colours)

Figure 7

Results of the parameter synthesis algorithm (II). The figure shows the second parameter (namely delay between the glucose signal and insulin secretion) of the glucose-insulin model whose value was discovered by our algorithm (see online version for colours)

Figure 8

Results of the parameter synthesis algorithm (III). The figure shows the third parameter (namely pancreatic responsivity to glucose) of the glucose-insulin model whose value was discovered by our algorithm (see online version for colours)

Figure 9

Exploring the parameter space. Our aim is to find a parameterised model that satisfies the null hypothesis H : p ≥ p₀. While exploring the parameter space, we move to the parameter that is closer to satisfying the null hypothesis (see online version for colours)

Figure 10

Our SPRT-based algorithm for parameter discovery. While exploring the parameter space, our algorithm uses the number of samples required by the SPRT (instead of calculating the actual probability at each point of the parameterised model satisfying the specification) in order to determine which parameter is better (see online version for colours)