(533e) Upscaled Model for Chemical Species Mass Transport in Homogeneous Regions of Cellular Systems

Studying mass transport and reaction of chemical species is relevant in biological systems as these establish the basis for the rates of cellular growth and metabolites production in industrial applications. Moreover, since cellular systems are hierarchical, a multiscale approach is required to properly predict the outcomes of specific system configurations and modes of culture. Few attempts to model species mass transport in cellular systems (mainly for biofilms) with a multiscale approach have been reported (Wood & Whitaker, 1999; Ginn et al., 2006; Kapellos et al., 2007; Orgogozo et al., 2010; Davit et al., 2010; Chabanon, 2015; De los Santos-Sánchez et al., 2016). Furthermore, there is no upscaled model that considers the microscale compartmentalization of key reactions and restrictions to the transport of certain chemical species inside the cell, which requires 3D modeling. This is important as cells are not homogeneous in space and time, and metabolic rates depend, in many cases, on the volume fraction of specific organelles.

In this work, interconnected quasi-steady upscaled models for glucose, glucose 6-phosphate, pyruvate, oxygen, two types of storage compounds, and secondary metabolites are derived in the extracellular, intracellular (cytoplasm, including organelles with characteristic length < 1 mm) and in the larger organelles of a biological system by using the volume averaging method. This method requires adopting a set of starting assumptions to formulate mass transport and reaction models at the microscale and then the imposition of reasonable upscaling assumptions to derive the corresponding upscaled models. These chemical species were strategically selected to allow studying glucose metabolism in any aerobic cell up to ATP formation, broadly considering the different metabolic routes for glucose in the cell.

For glucose, only extracellular convective transport with heterogeneous first-order reaction was considered at the microscale. Hence, the upscaled model incorporates convection, dispersion, and effective reaction. In the case of glucose 6-phosphate, cytoplasmic convective transport with heterogeneous and multiple homogeneous reactions were assumed in the microscale formulation. The corresponding upscaled model for glucose 6-phosphate, besides including dispersion and convection, contains reactive terms that account for its consumption (to yield reserve compounds and pyruvate) and production (from glucose and reserve compounds). For pyruvate, cytoplasmic convective and mitochondrial diffusive transport with multiple homogeneous reactions were proposed in the microscale model.

The upscaled model consists of two equations, one for the average concentration in the cytosol and another one for the average concentration in mitochondria. Both equations involve co-diffusive-like terms and terms due to the homogeneous (consumption and production) reactions that couple them with the other chemical species. For oxygen, extracellular convective, cytoplasmic convective and mitochondrial diffusive transport with an homogeneous reaction in the mitochondria were incorporated in the governing equations at the microscale. To simplify the derivations, an equilibrium upscaled model was proposed to represent oxygen transport in the extracellular region and the cytosol. As for the pyruvate, the model incorporates co-diffusion-like terms and the interfacial mass transport resistances, whereas the average model for oxygen in the mitochondria does not include convection, yet the effects of interfacial mass transport and homogeneous reactions play a relevant role. Furthermore, two types of storage compounds were considered, one resulting from glucose 6-phosphate and the other one from pyruvate. The microscale equations for both types of storage compounds include intracellular convective transport with homogeneous reactions. The corresponding upscaled models include convection, dispersion, and reaction terms; the latter coupling these equations to the other chemical species. Finally, for secondary metabolites, the microscale model included extracellular convective and cytoplasmic convective transport with homogeneous reaction. The upscaled models consider the contributions from convection, co-dispersion (between the cytosol and the extracellular phase), interfacial mass transport resistances, and, only in the upscaled model in the cytoplasm, a production term due to pyruvate. All macroscale models were written in terms of effective medium coefficients, predicted from the solution of associated closure problems for each chemical species.

The numerical solution of a steady version of the upscaled models was performed on a periodic 3D geometry inspired in a yeast liquid culture with organelles volume fraction according to literature reports. The extracellular fluid was considered Newtonian, while the intracellular fluid is assumed to follow a Carreau model. Also, the cell membrane (and wall, if applicable) is modeled as an interface with structural resistance to momentum transport. Moreover, flow results from the application of a macroscopic pressure forcing the fluid in the horizontal direction of the extracellular fluid phase, mimicking the effect of external agitation during liquid yeast cultivation in either a flask or a bioreactor. Here the effect of the (extracellular and intracellular) Reynolds (and thus Péclet) and Damköhler (of the different chemical species reactions that take place in a cell) numbers over the prediction of each effective-medium coefficient and average species concentration was evaluated. The selected values for the Reynolds and Damköhler numbers evaluated were according to viable scenarios during liquid medium cultivation of cells. In this way, enabling comparison and analysis of the effects of cultivation parameters and cell capacity (reaction rates of the metabolism) on species mass transport outside and inside cells. Finally, the upscaled models were validated with direct numerical simulations.

To conclude, the upscaled model derived here allows a detailed analysis of the repercussions of culture flow conditions and biological variables upon the transport of key chemical species. Ultimately, this affects the cellular system’s ability to take up and metabolize glucose and thus produce energy for its growth and metabolite production. It must be noted that the generality of this upscaled model enables its application to specific cellular systems and culture flow conditions, which is the only prerequisite to meeting the established starting and upscaling assumptions.