Matching Models Across Abstraction Levels with Gaussian Processes

Abstract

Biological systems are often modelled at different levels of abstraction depending on the particular aims/resources of a study. Such different models often provide qualitatively concordant predictions over specific parametrisations, but it is generally unclear whether model predictions are quantitatively in agreement, and whether such agreement holds for different parametrisations. Here we present a generally applicable statistical machine learning methodology to automatically reconcile the predictions of different models across abstraction levels. Our approach is based on defining a correction map, a random function which modifies the output of a model in order to match the statistics of the output of a different model of the same system. We use two biological examples to give a proof-of-principle demonstration of the methodology, and discuss its advantages and potential further applications.

GC and GS gratefully acknowledge support from the European Research Council under grant MLCS306999. LB acknowledges partial support from the EU project QUANTICOL, 600708, and by FRA-UniTS. We thank Dimitris Milios for useful discussions and for providing us with the MATLAB for heteroscedastic regression.

A Appendix

All the code that replicate these analysis is available at the corresponding author’s webpage, and hosted on Github (repository GP-correction-maps).

1.1 A.1 Further Details on the Examples

The two models from Sect. 5.1 correspond to these systems of differential equations

which we solved in MATLAB with the ode45 routine with all parameters (InitialStep, MaxStep, RelTol and AbsTol) set to 0.01.

Concerning the Protein Translation Network (PTN) in Sect. 5.2, the set of reactions and their propensity functions that we can use to derive a Continuous Time Markov Chain model of the network are the following. Here $\varvec{x}$ denotes a generic state of the system and, for instance, $\varvec{x}_\mathsf{{mRNA}}{}$ the number of mRNA copies in $\varvec{x}$.

The reduced PTN model is a special of this reactions set where transcription and mRNA decay are omitted. In this case we used StochPy to simulate the models and generate the input data per regression – see http://stochpy.sourceforge.net/; data sampling exploits python parallelism to reduce execution times.

For regression, we used the Gaussian Processes for Machine Learning toolbox for fixed-variance regression, see http://www.gaussianprocess.org/gpml/code/matlab/doc/ and a custom implementation of the other forms of regression.

Fig. 8.

Data generated to compute the satisfaction probability of the linear logic formula $\eta _2$ in Eq. (9). For each model 100 independent simulations are used to estimate the expectation of the probability. The regression input space is the same used to compute $\eta _1$, but the models are simulated with just one inactive gene in the initial state. The heteroscedastic variance in the regression is computed as the variance of the correction of the expected satisfaction probability (point-wise $\sigma$-estimator); the fixed-variance regression is computed by estimating the variance from the data (empirical $\overline{\sigma }$-estimator).

1.2 A.2 Proofs

Proof of Theorem 1

Proof

Both the empiricals and nested estimator rely on an unbiased estimator of the mean/variance, which means that if $k\rightarrow \infty$, i.e., we sample all possible values for the free variables, we would have a true model of $\overline{y}$ $\sigma$. This means that, for each sampled value from $\varTheta$, even the simplest $\overline{\sigma }$-estimator would be equivalent, in expectation, to the marginalization of the free variables. This is enough, combined with properties of Gaussian Processes regression (i.e., the convergence to the true model with infinite training points), to state that the overall approach leads to an unbiased estimator of the correction map.