## Abstract

Identifying feasible steady state solutions of a brain energy metabolism model is an inverse problem that allows infinitely many solutions. The characterization of the non-uniqueness, or the uncertainty quantification of the flux balance analysis, is tantamount to identifying the degrees of freedom of the solution. The degrees of freedom of multi-compartment mathematical models for energy metabolism of a neuron–astrocyte complex may offer a key to understand the different ways in which the energetic needs of the brain are met. In this paper we study the uncertainty in the solution, using techniques of linear algebra to identify the degrees of freedom in a lumped model, and Markov chain Monte Carlo methods in its extension to a spatially distributed case. The interpretation of the degrees of freedom in metabolic terms, more specifically, glucose and oxygen partitioning, is then leveraged to derive constraints on the free parameters to guarantee that the model is energetically feasible. We demonstrate how the model can be used to estimate the stoichiometric energy needs of the cells as well as the household energy based on the measured oxidative cerebral metabolic rate of glucose and glutamate cycling. Moreover, our analysis shows that in the lumped model the net direction of lactate dehydrogenase (LDH) in the cells can be deduced from the glucose partitioning between the compartments. The extension of the lumped model to a spatially distributed multi-compartment setting that includes diffusion fluxes from capillary to tissue increases the number of degrees of freedom, requiring the use of statistical sampling techniques. The analysis of the distributed model reveals that some of the conclusions valid for the spatially lumped model, e.g., concerning the LDH activity and glucose partitioning, may no longer hold.

Original language | English (US) |
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Pages (from-to) | 1823-1849 |

Number of pages | 27 |

Journal | Journal of Mathematical Biology |

Volume | 73 |

Issue number | 6-7 |

DOIs | |

State | Published - Dec 1 2016 |

Externally published | Yes |

### Bibliographical note

Funding Information:The work of Daniela Calvetti was partly supported by Grant Number 246665 from the Simons Foundation, and the work of Erkki Somersalo was partly supported by NSF Grant DMS 1016183. Daniela Calvetti and Erkki Somersalo were partly supported by NIH, grant 1U01GM111251-01.

Publisher Copyright:

© 2016, Springer-Verlag Berlin Heidelberg.

## Keywords

- Bayesian flux balance analysis
- Brain energy metabolism
- Distributed model
- Glucose partitioning
- Lactate shuttle