Optimization and model reduction in the high dimensional parameter space of a budding yeast cell cycle model

Oguz et al. BMC Systems Biology 2013, 7:53 http://www.biomedcentral.com/1752-0509/7/53 RESEARCH ARTICLE Open Access Optimization and model reduction in the high dimensional parameter space of a budding yeast cell cycle model Cihan Oguz1 , Teeraphan Laomettachit2 , Katherine C Chen1 , Layne T Watson3 , William T Baumann4 and John J Tyson1* Abstract Background: Parameter estimation from experimental data is critical for mathematical modeling of protein regulatory networks. For realistic networks with dozens of species and reactions, parameter estimation is an especially challenging task. In this study, we present an approach for parameter estimation that is effective in fitting a model of the budding yeast cell cycle (comprising 26 nonlinear ordinary differential equations containing 126 rate constants) to the experimentally observed phenotypes (viable or inviable) of 119 genetic strains carrying mutations of cell cycle genes. Results: Starting from an initial guess of the parameter values, which correctly captures the phenotypes of only 72 genetic strains, our parameter estimation algorithm quickly improves the success rate of the model to 105–111 of the 119 strains. This success rate is comparable to the best values achieved by a skilled modeler manually choosing parameters over many weeks. The algorithm combines two search and optimization strategies. First, we use Latin hypercube sampling to explore a region surrounding the initial guess. From these samples, we choose ∼20 different sets of parameter values that correctly capture wild type viability. These sets form the starting generation of differential evolution that selects new parameter values that perform better in terms of their success rate in capturing phenotypes. In addition to producing highly successful combinations of parameter values, we analyze the results to determine the parameters that are most critical for matching experimental outcomes and the most competitive strains whose correct outcome with a given parameter vector forces numerous other strains to have incorrect outcomes. These “most critical parameters” and “most competitive strains” provide biological insights into the model. Conversely, the “least critical parameters” and “least competitive strains” suggest ways to reduce the computational complexity of the optimization. Conclusions: Our approach proves to be a useful tool to help systems biologists fit complex dynamical models to large experimental datasets. In the process of fitting the model to the data, the tool identifies suggestive correlations among aspects of the model and the data. Keywords: Optimization, Budding Yeast, Cell Cycle, ODE Model, Model Reduction, Phenotypic Constraints, Latin Hypercube Sampling, Differential Evolution, Sensitivity Analysis, Phenotype Competition Background The challenges facing molecular systems biologists include the development of accurate mathematical models of complex biological processes [1], the elucidation of design principles that control biological behavior [2], *Correspondence: 1 Department of Biological Sciences, Virginia Tech, Blacksburg, Virginia 24061, USA Full list of author information is available at the end of the article and the generation of new insights into biology that are not apparent solely from experimental studies [3]. A common mathematical method to address these challenges is dynamical systems theory [4,5], the use of nonlinear ordinary differential equations (ODEs) to describe the way networks of biochemical reactions change in time. By comparing the temporal development of the model under conditions that simulate a variety of experimental protocols with the observed behavior of the biological system © 2013 Oguz et al.; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Oguz et al. BMC Systems Biology 2013, 7:53 http://www.biomedcentral.com/1752-0509/7/53 under the same conditions, one can evaluate how well or poorly the mathematical model performs. Our focus in this study is parameter estimation of a nonlinear and high-dimensional ODE model (> 100 model parameters) that is constrained by a large number of dissimilar experimental observations. The non-differentiable nature of our objective function (described in the next section) led to our choice of a stochastic global optimization approach [6,7] that relies on an evolutionary search, namely differential evolution (DE) [8], starting from a diverse population of parameter vectors scattered over a feasible region of parameter space. DE is a popular global optimization method due to its efficiency and simplicity. However, we should mention that a recent novel (yet more complex) algorithm outperformed DE in multiple optimization tasks with large scale systems biology models due to extensive local search capability [9] that is lacking in the simplest form of DE. For a recent comprehensive review regarding the application of DE and other metaheuristic optimization techniques in systems biology, we refer the reader to [7]. Parameter estimation is not only about finding an “optimal” set of parameter values for fitting a collection of experimental observations. During the course of the global optimization procedure, we expect to find many different parameter vectors that do equally well (or nearly as well) as the best one. Working with this sample of “quite good” sets of parameter values, we can quantify how well the experimental data constrain individual parameter values. We can distinguish critical parameters (highly constrained by the data) from irrelevant parameters (those that have little bearing on optimization of the objective function) [10]. We can distinguish those experimental results that provide the most information about the underlying model from those that provide the least, and we can design new experiments that will provide the most new information about the underlying molecular regulatory system [11-13]. All these types of information can be very useful in refining and extending the model [14]. Our research group has been interested for many years in the molecular mechanisms controlling the cell division cycle of budding yeast. The main events of the cell cycle (DNA synthesis and mitosis) are controlled in budding yeast, and indeed in all eukaryotic cells, by a family of protein kinases called cyclin-dependent kinases (CDKs) [15]. We have built comprehensive and accurate models of the periodic activation of CDKs, based on nonlinear ODEs describing the underlying biochemical reaction network [16]. The models are used to understand how CDKs control cell cycle progression in normal (“wild type”) yeast cells, and also how cell cycle progression is altered in yeast strains harboring mutations in genes of the CDK (...truncated)