Optimal experiment design for model selection in biochemical networks
© Vanlier et al.; licensee BioMed Central Ltd. 2014
Received: 11 April 2013
Accepted: 13 February 2014
Published: 20 February 2014
Mathematical modeling is often used to formalize hypotheses on how a biochemical network operates by discriminating between competing models. Bayesian model selection offers a way to determine the amount of evidence that data provides to support one model over the other while favoring simple models. In practice, the amount of experimental data is often insufficient to make a clear distinction between competing models. Often one would like to perform a new experiment which would discriminate between competing hypotheses.
We developed a novel method to perform Optimal Experiment Design to predict which experiments would most effectively allow model selection. A Bayesian approach is applied to infer model parameter distributions. These distributions are sampled and used to simulate from multivariate predictive densities. The method is based on a k-Nearest Neighbor estimate of the Jensen Shannon divergence between the multivariate predictive densities of competing models.
We show that the method successfully uses predictive differences to enable model selection by applying it to several test cases. Because the design criterion is based on predictive distributions, which can be computed for a wide range of model quantities, the approach is very flexible. The method reveals specific combinations of experiments which improve discriminability even in cases where data is scarce. The proposed approach can be used in conjunction with existing Bayesian methodologies where (approximate) posteriors have been determined, making use of relations that exist within the inferred posteriors.
Previously, predictive distributions have been used to perform experiment design targeted at reducing the uncertainty of specific predictions [29–31]. In the field of machine learning, optimal experiment design based on information-theoretic considerations is typically referred to as active learning . In the neurosciences, the Bayesian inversion and selection of nonlinear states space models is known as dynamic causal modelling (DCM). Although DCM is dominated by variational (approximate) Bayesian model inversion - the basic problems and ensuing model selection issues are identical to the issues considered in this work. In DCM, the issue of optimising experimental design focuses on the Laplace-Chernoff risk for model selection and its relationship with classical design optimality criteria. Daunizeau et al. (2011) consider the problem of detecting feedback connections in neuronal networks and how this depends upon the duration of design stimulation . We will consider a similar problem in biochemical networks - in terms of identifying molecular interactions and when to sample data. We present a method to use samples from simulated predictive distributions for selecting experiments useful for model selection. Considering the increased use of Bayesian inference in the field [14, 34–39], this approach is particularly timely since it enables investigators to extract additional information from their inferences.
There are many design parameters that one could optimize. In this paper, we focus on a simple example: namely, which system variable should be measured and at which time point. We argue that by measuring those time points at which the models show the largest difference in their predictive distributions, large improvements in the Bayes factors can be obtained. By applying the methodology on an analytically tractable model, we show that the JSD is nearly monotonically related to the predicted change in Bayes factor. Subsequently, the Jensen Shannon divergence is computed between predictions of a non-linear biochemical network. Since each model implies different relations between the predictive distributions, certain combinations of predictions lead to more discriminability than others. The method serves as a good predictor for effective experiments when compared to the obtained Bayes factors after the measurements have been performed. The approach can be used to design multiple experiments simultaneously, revealing benefits that arise from combinations of experiments.
Consider biochemical networks that can be modeled using a system of ordinary differential equations. These models comprise of equations which contain parameters (constant in time), inputs and state variables . Given a set of parameters, inputs, and initial conditions , these equations can be simulated. Measurements are performed on a subset and/or a combination of the total number of state variables in the model. Measurements are hampered by measurement noise , while many techniques used in biology necessitate the use of scaling and offset parameters . The vector , defined as , lists all the required quantities to simulate the model. The parameters determine the experimental design and could include differences in when various responses are measured or the mapping from hidden model states to observed responses . We will refer to these as ‘design parameters’ that are, crucially, distinguished from model parameters . Design parameters are under experimental control and determine the experimental design. In what follows, we try to optimise the discriminability of models in terms of Bayesian model comparison by optimizing an objective function with respect to . In the examples, we will consider as the timing of extra observations.
where M i indicates a model and y D the observed data. The parameters are given by , while indicates the value of a data point of state variable k at time j, respectively.
where is the distribution of the observed data given parameters . The parameter prior distributions typically reflect the prior uncertainty associated with the parameters. To sample from the posterior parameter distribution, one needs to verify that the posterior distribution is proper. This can be checked by profiling the different parameters and determining whether the likelihood times the prior does not flatten out [28, 46]. After checking whether the posterior distribution of parameters is proper, a sample from this distribution can be obtained using Markov Chain Monte Carlo (MCMC) [22, 28]. MCMC can generate samples from probability distributions whose probability densities are known up to a normalizing factor  (see Additional file 1: section S1). A sample of the posterior parameter distribution reflects the uncertainty associated with the parameter values and can subsequently be used to simulate different predictions. The predictive distribution can now be sampled by simulating the model for each of the samples in the posterior parameter distribution and adding noise generated by the associated error model. The latter is required as future observations will also be affected by noise.
where M 1 and M 2 refer to the different models under consideration. Unnecessarily complex models are implicitly penalized due to the fact that these result in a lower weighting of the high likelihood region, which results in a lower value for the integrated likelihood. This is illustrated in Figure 2. This means that maximizing the model evidence corresponds to maintaining an accurate explanation for the data while minimizing complexity .
Bounds can be defined where the Bayes factor value becomes decisive for one model over the other. Typically, a ratio of 100:1 is considered decisive in terms of model selection [40, 51]. In dynamic causal modelling, variational methods predominate, usually under the Laplace assumption. This assumes that the posterior density has a Gaussian shape, which greatly simplifies both the integration problems and numerics. Note that assuming a Gaussian posterior over the parameters does not necessarily mean that the posterior predictive distribution over the data is Gaussian (see Figure 1). Computing the required marginal likelihoods is challenging for non-linear problems where such asymptotic approximations to the posterior distribution are not appropriate. Here one must resort to more advanced methods such as thermodynamic integration (see Additional file 1: section S2)  or annealed importance sampling . Though the Bayes factor is a useful method of model selection, determining what to measure in order to improve the Bayes factor in favor of the correct model is a non-trivial problem. As such, it provides a means to perform model selection, but not the optimal selection of data features.
Here P and Q are random variables with p and q their associated densities. Considering that only a sample of the PPDs is available, it is required to obtain a density estimate suitable for integration. Density estimation can be approached in two ways: by Kernel Density Estimation (KDE), or by k-Nearest Neighbor (kNN) density estimation. In Kernel Density Estimation (KDE), an estimate of the density is made by centering normalized kernels on each sample and computing weighted averages. This results in a density estimate with which computations can be performed. The kernels typically contain a bandwidth parameter which is estimated by means of cross validation [54, 55].
For well behaved low dimensional distributions, KDE often performs well. Considering the strongly non-linear nature of both the parameter and predictive distributions, a Gaussian kernel with constant covariance is not appropriate. As the dimensionality and non-uniformity of the problem increases, more and more weights in the KDE become small and estimation accuracy is negatively affected . Additionally, choosing an appropriate bandwidth by means of cross-validation is a computationally expensive procedure to perform for each experimental candidate.
Testing the method: numerical experiments
To demonstrate the method, a series of simulation studies are performed. Since we know which model generated the data, it is possible to compare to the Bayes factor pointing to the true model. After generating an initial data set using the true model, PPDs are sampled for each of the competing models. As the design variable, we consider the timing of a new measurement. Hence, we look at differences between the predictive distributions belonging to the different models at different timepoints. We use a sample of simulated observables at specific timepoints to compute JSD estimates between the different models. We thereby test whether the JSD estimate can be used to compare different potential experiments. The new experimental data is subsequently included and the JSD compared to the change in Bayes factor in favor of the correct model. Note that this new Bayes factor depends on the experimental outcome and that this approach results in a distribution of predicted Bayes factors. A large change in Bayes factor indicates a useful experiment.
where represents a parameter vector and B constitutes a design matrix with basis functions B i (t). Given that σ, the standard deviation of the Gaussian observation error ε, is known and the prior distribution over the parameters is a Gaussian with standard deviation ξ, the mean and covariance matrix of the posterior distribution can be computed analytically (see Additional file 1: section S4). Using linear models avoids the difficult numerical integration commonly required to compute the Bayes factor and makes it possible to perform overarching Monte Carlo studies on how these Bayes factors adjust upon including new experimental data. The analytical expressions make it possible to compare the JSD to distributions of the actual Bayes factors for model selection.
Non-linear biochemical networks
Each of the artificial models was able to describe the measured data to an acceptable degree. We used a Gamma distribution with α=1 and β=3 for the prior distributions of the parameters. This prior is relatively non-informative (allowing a large range of parameter values), while not being so vague that the simplest model is always preferred (Lindley’s paradox ). Data was obtained using M 1. Observables were Bp, of which three replicates were measured, and Dp, of which two replicates were measured. These were measured at t=[ 0,2,5,10,20,40,60,100]. All replicates were simulated by adding Gaussian white noise with a standard deviation of 0.03. The parameter values corresponding to the true system were obtained by running Monte Carlo simulations until a visible overshoot above the noise level was observed. Parameter inference was performed using population MCMC with the noise σ as an inferred parameter. As design variables we consider both the choice of which observable(s) to measure and the time point(s) of the measurement.
All algorithms were implemented in Matlab (Natick, MA). Numerical integration of the differential equations was performed with compiled MEX files using numerical integrators from the SUNDIALS CVode package (Lawrence Livermore National Laboratory, Livermore, CA). Absolute and relative tolerances were set to 10−8 and 10−9 respectively. MCMC was performed using a population MCMC approach using N T =40 chains with a temperature schedule given by . This also permitted computation of the Bayes factors between the non-linear models by means of thermodynamic integration. The Gaussian proposal distribution for the MCMC was based on an approximation to the Hessian computed using a Jacobian obtained by simulating the sensitivity equations. After convergence, the chain was thinned to 10000 samples. Since the number of experiments designed simultaneously (and therefore the dimensionality of the prediction vectors) was reasonably small (N samples >>2 k ), the kNN search was performed using k-d trees . The figures in this paper were determined using k=10.
Results and discussion
1. Steady state Cp and BpCp concentration
2. Bp and Dp during the peak in the second condition (u=2)
3. Steady state Cp
Sample table title
Δ B 12
Δ B 13
Δ B 14
This paper describes a method applicable to performing experiment design with the aim of differentiating between various hypotheses. We show by means of a simulation study on analytically tractable models that the JSD is approximately monotonically related to the expected change in Bayes factor in favor of the model that generated the data (considering the current uncertainty in its parameters). This monotonic relation is useful, because it implies that the JSD can be used as a predictor of the change in Bayes factor. The applicability to non-linear models of biochemical reaction networks was demonstrated by applying it to models based on motifs previously observed in signaling networks [58, 59]. Experiments were designed for distinguishing between different feedback mechanisms.
Though forecasting a predictive distribution of Bayes factors has been suggested , the implicit penalization of model complexity could have adverse consequences. The experiment design could suggest a measurement where the probability densities of two models overlap. When this happens, both models can describe the data equally well, which leads to an implicit penalization of the more complex model (since it allows for more varied predictions due to its added freedom). This penalization can then be followed by subsequent selection (of the simpler model). Though a decisive selection occurs, such an experiment would not provide additional insight however. In , this is mitigated by determining the evidence in favor of a more complex model. Moreover, computing the predictive distributions of Bayes factors required for this approach is computationally intractable for non-linear models that are not nested. By focusing on differences in predictive distributions, both these problems are mitigated, making it is possible to pinpoint where the different models predict different behavior. Aside from their usefulness in model selection, such predictive differences could also be attributed to the different mechanisms present in the different models. This allows for follow-up studies to investigate whether these are either artificial or true system behavior.
A complicating factor in this method is the computational burden. The largest challenge to overcome is to obtain a sample from the posterior parameter distribution. Running MCMC on high dimensional problems can be difficult. Fortunately, recent advances in both MCMC [19, 62] as well as approximate sampling techniques [39, 48, 63, 64] allow sampling parameter distributions of increasingly complex models [14, 34–38]. The bottleneck in computing the JSD resides in searching for the k th nearest neighbor. A subproblem which occurs in many different problems and for which computationally faster solutions exist [65, 66]. An attractive aspect of this methodology is that it is possible to design multiple experiments at once. However, the density estimates typically become less accurate as the number of designed experiments increases (see Additional file 1: section S8). Therefore, we recommend starting with a low number of experiments (two or three) and gradually adding experiments while the JSD is low. Density estimation can also be problematic when the predictions vary greatly in their dispersion. When considering non-negative quantities such as concentrations, log-transforming the predictions may alleviate problems. Finally, the number of potential combinations of experiments increases exponentially with the number of experiments designed. It is clear that this rapidly becomes infeasible for large numbers of experiments. However, it is not necessary to fill the entire experimental matrix and techniques such as Sequential Monte Carlo sampling could be considered as an alternative to more effectively probe this space. We revert the reader to Additional file 1: section S7 for a proof of principle implementation of such a sampler.
One additional point of debate is the weighting of each of the models in the mixture distribution used to compute the JSD. It could be argued that it would be more sensible to weight models according to their model probabilities by determining the integrated likelihoods of the data that is already available. The reason for not doing this is two-fold. Firstly, the computational burden this adds to the experimental design procedure is significant. More importantly however, the implicit weighting in favor of parsimony could strongly affect the design by removing models which are considered unnecessarily complex at this stage of the analysis. When designing new experiments, the aim is to obtain measurements that allow for optimal discrimination between the predictive distributions under the different hypotheses. Optimal discrimination makes it sensible to consider the models equally probable a priori.
The method has several advantages that are particularly useful for modeling biochemical networks. Because the method is based on sampling from the posterior parameter probability distribution, it is particularly suitable when insufficient data is available to consider Gaussian parameter probability distributions or model linearisations. Additionally, it allows incorporation of prior knowledge in the form of prior parameter probability distributions. This is useful when the available data contains insufficient constraints to result in a well defined posterior parameter distribution. Because the design criterion is based on predictive distributions and such distributions can be computed for a wide range of model quantities, the approach is very flexible. In biochemical research, in vivo measurements are often difficult to perform and practical limitations of the various measurement technologies play an important role. In many cases measurements on separate components cannot be performed and measurements result in indirect derived quantities. Fortunately, in the current framework such measurements can be used directly since distributions of such experiments can be predicted.
Moreover, the impact of specific combinations of experiments can be assessed by including them in the design simultaneously which reveals specific combination of measurements that are particularly useful. This way, informative experiments can be distinguished from non-informative ones and the experimental efforts can be targeted to discriminate between competing hypotheses.
Source code is available at: http://cbio.bmt.tue.nl/sysbio/software/pua.html.
This work was funded by the Netherlands Consortium for Systems Biology (NCSB). The authors would like to thank H. W. H. van Roekel, R. M. Foster and C. Çölmekçi Öncü for helpful discussions.
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