際際滷

際際滷Share a Scribd company logo

Poster for Information, probability and inference in systems biology (IPISB 2013)

C
Colin Gillespie

Interest lies in inference for the rate parameters in a complex stochastic biological model describing the aggregation of proteins within human cells. Protein aggregation is a factor in many age-related diseases such as Alzheimer's disease. Ideally time-course measurements on all chemical species in the model would be available. However, current experimental techniques only allow noisy observations on the proportions of cell death at a few time points. Although the model has a large state space and is analytically intractable, realisations from the model can be obtained using a stochastic simulator. The time evolution of a cell can be repeatedly simulated giving an estimate of the proportion of cell death. Unfortunately, simulation from the model is too slow to be used in an MCMC inference scheme. A Gaussian process emulator, which is very fast, can be used to approximate the simulator. An MCMC scheme can be constructed targeting the posterior distribution of interest, however evaluating the marginal likelihood is challenging. A pseudo-marginal approach replaces the marginal likelihood with an easy to construct unbiased estimate while still targeting the true posterior. The methods will be illustrated using a toy birth-death model, allowing comparison with the exact model.

1 of 1
Download to read offline
Inferring parameters in a large stochastic model using only proportions of cell death: insights from the birth-death model Holly Ainsworth, Richard Boys & Colin Gillespie Newcastle University, UK REFERENCES [1] Tang, M.Y., Proctor, C.J., Woulfe, J., Gray, D.A. Experimental and Computational Analysis of Polyglutamine-Mediated Cytotoxicity In PLoS Com- put Biol., 2010 . INTRODUCTION Expanded polyglutamine (PolyQ) proteins are known to be the causative agents in a number of neurodegenerative diseases, but they are still poorly understood. Aggregation of speci鍖c proteins are part of the normal process of ageing in the brain as well as in many age-related diseases. A causative link between aggregation and disease is not universally accepted. STOCHASTIC MODEL Stochastic kinetic model with uncertain pa- rameters: 27 (chemical) species 70 reactions 40 rate constants, denoted 慮 The model aims to explore the relationship between PolyQ, p38MAPK activation, genera- tion of reactive oxygen species (ROS), protea- some inhibition and inclusion body formation. DATA Data are proportions (of cell death) - not quan- titative trait measurements Scenario 24hrs 36hrs 48hrs GFP 15.03 14.55 26.08 H25 18.97 18.07 22.50 H103 21.68 23.44 36.44 SIMULATION Denote the probability of cell death at time t, pt(慮). Note the probability of cell death depends on parameters 慮. Given 慮, the Gillespie algorithm can be used to simulate the time evolution of a particular cell: The simulation gives us a binary time series for the cell, with 1 = death and 0 = no death. Repeating the above for n cells gives us a handle on pt(慮) via the observed proportion of cell death pt(慮), where pt(慮) 1 n Bin(n, pt(慮)). BIRTH-DEATH MODEL Let x denote the number of individuals present in the population. In chemical kinetic notation, this system is represented as R1 : x 了 2x (birth) R2 : 2x 袖 x (death) Example simulations from the model: q qq0 5 10 15 0 4 8 12 Time Population 0.00 0.25 0.50 0.75 1.00 0 4 8 12 Time Probabilityofextinction Simulator n = 10 n = 100 n = 1000 n = 10000 Comparisons with PolyQ model: Compare a single population governed by a birth-death process with a cell governed by the PolyQ model. A cell becoming extinct in the birth-death process can be likened to a cell dying in the PolyQ model. Proportions of extinction from the birth- death model are comparible to propor- tions of cell death from the PolyQ model. An analytic expression for the probability of extinction in the birth-death process for given t, 了, 袖 and initial population level is available. INFERENCE Work with proportions on the logit scale and assume data model: yt = logit xt = logit pt(慮) + t, t = 1, . . . , T where t N(0, 1) independently. The posterior of interest is (慮, |y) (慮)()(y|慮, ). Approaches to inference: Vanilla MCMC - approximate the distribution of pt(慮) and account for the uncertainty using elogit pt(慮) N logit pt(慮), 1 npt(慮)[1 pt(慮)] approximately. Pseudo marginal 1 - construct a Monte Carlo estimate of the marginal likelihood. Pseudo marginal 2 - at each iteration of the MCMC scheme, use a (SIR) particle 鍖lter to construct a SMC approximation to the marginal likelihood. RESULTS log(了) log(袖) log() 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 n=10n=100n=1000 3 2 1 0 1 0.5 0.0 0.5 1.0 3 2 1 0 1 Parameter value Density Vanilla MCMC Pseudomarginal MCMC 1 Pseudomarginal MCMC 2 Inference using exact probability of death Both pseudo-marginal schemes involve more work. They use n#particles runs of the simulator at each iteration, com- pared to n runs for the original scheme. The pseudo-marginal approach has the advantage that it performs exact infer- ence (for a particular choice of n). This will be useful particularly when the asymptotic distributional result for elogit pt(慮) is poor for small n. The SMC approach to estimating the marginal likelihood appears to mix better than the MC approach.

More Related Content

Poster for Information, probability and inference in systems biology (IPISB 2013)

  • 1. Inferring parameters in a large stochastic model using only proportions of cell death: insights from the birth-death model Holly Ainsworth, Richard Boys & Colin Gillespie Newcastle University, UK REFERENCES [1] Tang, M.Y., Proctor, C.J., Woulfe, J., Gray, D.A. Experimental and Computational Analysis of Polyglutamine-Mediated Cytotoxicity In PLoS Com- put Biol., 2010 . INTRODUCTION Expanded polyglutamine (PolyQ) proteins are known to be the causative agents in a number of neurodegenerative diseases, but they are still poorly understood. Aggregation of speci鍖c proteins are part of the normal process of ageing in the brain as well as in many age-related diseases. A causative link between aggregation and disease is not universally accepted. STOCHASTIC MODEL Stochastic kinetic model with uncertain pa- rameters: 27 (chemical) species 70 reactions 40 rate constants, denoted 慮 The model aims to explore the relationship between PolyQ, p38MAPK activation, genera- tion of reactive oxygen species (ROS), protea- some inhibition and inclusion body formation. DATA Data are proportions (of cell death) - not quan- titative trait measurements Scenario 24hrs 36hrs 48hrs GFP 15.03 14.55 26.08 H25 18.97 18.07 22.50 H103 21.68 23.44 36.44 SIMULATION Denote the probability of cell death at time t, pt(慮). Note the probability of cell death depends on parameters 慮. Given 慮, the Gillespie algorithm can be used to simulate the time evolution of a particular cell: The simulation gives us a binary time series for the cell, with 1 = death and 0 = no death. Repeating the above for n cells gives us a handle on pt(慮) via the observed proportion of cell death pt(慮), where pt(慮) 1 n Bin(n, pt(慮)). BIRTH-DEATH MODEL Let x denote the number of individuals present in the population. In chemical kinetic notation, this system is represented as R1 : x 了 2x (birth) R2 : 2x 袖 x (death) Example simulations from the model: q qq0 5 10 15 0 4 8 12 Time Population 0.00 0.25 0.50 0.75 1.00 0 4 8 12 Time Probabilityofextinction Simulator n = 10 n = 100 n = 1000 n = 10000 Comparisons with PolyQ model: Compare a single population governed by a birth-death process with a cell governed by the PolyQ model. A cell becoming extinct in the birth-death process can be likened to a cell dying in the PolyQ model. Proportions of extinction from the birth- death model are comparible to propor- tions of cell death from the PolyQ model. An analytic expression for the probability of extinction in the birth-death process for given t, 了, 袖 and initial population level is available. INFERENCE Work with proportions on the logit scale and assume data model: yt = logit xt = logit pt(慮) + t, t = 1, . . . , T where t N(0, 1) independently. The posterior of interest is (慮, |y) (慮)()(y|慮, ). Approaches to inference: Vanilla MCMC - approximate the distribution of pt(慮) and account for the uncertainty using elogit pt(慮) N logit pt(慮), 1 npt(慮)[1 pt(慮)] approximately. Pseudo marginal 1 - construct a Monte Carlo estimate of the marginal likelihood. Pseudo marginal 2 - at each iteration of the MCMC scheme, use a (SIR) particle 鍖lter to construct a SMC approximation to the marginal likelihood. RESULTS log(了) log(袖) log() 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 n=10n=100n=1000 3 2 1 0 1 0.5 0.0 0.5 1.0 3 2 1 0 1 Parameter value Density Vanilla MCMC Pseudomarginal MCMC 1 Pseudomarginal MCMC 2 Inference using exact probability of death Both pseudo-marginal schemes involve more work. They use n#particles runs of the simulator at each iteration, com- pared to n runs for the original scheme. The pseudo-marginal approach has the advantage that it performs exact infer- ence (for a particular choice of n). This will be useful particularly when the asymptotic distributional result for elogit pt(慮) is poor for small n. The SMC approach to estimating the marginal likelihood appears to mix better than the MC approach.