Fitting and comparison of calcium-calmodulin kinetic schemes to a common data set using non-linear mixed effects modelling

Abstract

Calmodulin is a calcium binding protein that is essential in calcium signalling in the brain. There are many computational models of calcium-calmodulin binding that capture various calmodulin features. However, existing models have generally been fit to different data sets, with some publications not reporting their training and validation performance. Moreover, there is no model comparison using a common benchmark data set as is common practice in other modeling domains. Finally, some calmodulin models have been fit as a part of a larger kinetic scheme, which may have resulted in parameters being underdetermined. We address these three limitations of previous models by fitting the published calcium-calmodulin schemes to a common calcium-calmodulin data set comprising of equlibrium data from Shifman et al. and dynamical data from Faas et al. Due to technical limitations, the amount of uncaged calcium in Faas et al. data could not be predicted with certainty. To find good parameter fits, despite this uncertainty, we used non-linear mixed effects modelling as implemented in the Pumas.jl package. The Akaike information criterion values for our reaction rates were significantly lower than for the published parameters, indicating that the published parameters are suboptimal. Moreover, there were significant differences in calmodulin activation, both between the schemes and between our reaction rate and those previously published. A kinetic scheme with independent lobes and unique, rather than identical, binding sites fit the data best. Our results support two hypotheses: (1) partially bound calmodulin is important in cellular signalling; (2) calcium binding sites within a calmodulin lobe are kinetically distinct rather than identical. We conclude that more attention should be given to validation and comparison of models of individual molecules. Author summaryLearning and memory depend on changes in synapses, the connections between nerve cells. To understand how learning and memory work, it is important to understand how the proteins involved in these changes are regulated. Computational modelling of biochemical reactions has been used to understand the regulation processes involved in learning and memory. However, computational models often rely on simplifications of biochemical reaction schemes, and it can be difficult to tell which simplifications are "the best", i.e. capture important aspects of a proteins behaviour. It is also difficult to estimate the best model parameters, such as binding reaction rates or binding strengths. Modellers often base their estimates on known experimental results, but often not in a structured way. We examine and compare a variety of computational models of calmodulin, a protein necessary for changes in synaptic strength. We apply a new approach to infer the model parameters that best fit two published datasets, and compare our parameters to previously published ones. We find that some simplified reaction schemes are more successful than others in capturing the activation pattern of calmodulin. We also find that the parameter sets found through our approach outperform previously published parameter sets in fitting the experimental data.