Abstract
BackgroundStochastic fluctuations in molecular numbers have been in many cases shown to be crucial for the understanding of biochemical systems. However, the systematic study of these fluctuations is severely hindered by the high computational demand of stochastic simulation algorithms. This is particularly problematic when, as is often the case, some or many model parameters are not well known. Here, we propose a solution to this problem, namely a combination of the linear noise approximation with optimisation methods. The linear noise approximation is used to efficiently estimate the covariances of particle numbers in the system. Combining it with optimisation methods in a closed-loop to find extrema of covariances within a possibly high-dimensional parameter space allows us to answer various questions. Examples are, what is the lowest amplitude of stochastic fluctuations possible within given parameter ranges? Or, which specific changes of parameter values lead to the increase of the correlation between certain chemical species? Unlike stochastic simulation methods, this has no requirement for small numbers of molecules and thus can be applied to cases where stochastic simulation is prohibitive.ResultsWe implemented our strategy in the software COPASI and show its applicability on two different models of mitogen-activated kinases (MAPK) signalling -- one generic model of extracellular signal-regulated kinases (ERK) and one model of signalling via p38 MAPK. Using our method we were able to quickly find local maxima of covariances between particle numbers in the ERK model depending on the activities of phospho-MKKK and its corresponding phosphatase. With the p38 MAPK model our method was able to efficiently find conditions under which the coefficient of variation of the output of the signalling system, namely the particle number of Hsp27, could be minimised. We also investigated correlations between the two parallel signalling branches (MKK3 and MKK6) in this model.ConclusionsOur strategy is a practical method for the efficient investigation of fluctuations in biochemical models even when some or many of the model parameters have not yet been fully characterised.
Highlights
Stochastic fluctuations in molecular numbers have been in many cases shown to be crucial for the understanding of biochemical systems
We demonstrate the application of this new strategy on two different models of mitogen-activated kinase (MAPK) signalling pathways, namely a model of extracellular signal-regulated kinases (ERK) by Kholodenko [23] and a model of p38 mitogen-activated protein kinases (MAPK) by Hendriks et al [24]
Implementation of the method in COPASI The software COPASI [21,22] gives all interested researchers easy access to modelling and simulation for biochemical networks, because it is freely available under the Artistic license version 2.0 at [22] and supports the Systems Biology Markup Language (SBML) standard [25] for the exchange of model files with other software
Summary
Stochastic fluctuations in molecular numbers have been in many cases shown to be crucial for the understanding of biochemical systems. The systematic study of these fluctuations is severely hindered by the high computational demand of stochastic simulation algorithms This is problematic when, as is often the case, some or many model parameters are not well known. The linear noise approximation is used to efficiently estimate the covariances of particle numbers in the system Combining it with optimisation methods in a closed-loop to find extrema of covariances within a possibly high-dimensional parameter space allows us to answer various questions. Different mathematical formalisms have been developed to allow stochastic modelling and to explicitly take into account random fluctuations Such systems are usually modelled by a continuous-time Markov process which follows the chemical master equation. The calculation of very many of them quickly becomes impracticable even when accelerated approximate stochastic simulation methods [8] are employed
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