Jessica Sánchez-Vargas
email: jessica.sanchezvarg@gmail.com

Francisco J Valdés-Parada

Didier Lasseux

Mauricio A Trujillo-Roldán

EFFECTIVE MEDIUM MODEL OF HYPHAL FLUID MECHANICS

Upscaled mathematical models for mass and momentum transport inside fungal pellets were derived and used to predict the average intracellular and extracellular velocity and permeability of pellets of Laccaria trichodermophora.

HOW TO MODEL FLUID MECHANICS IN A FUNGAL PELLET? HOW TO MODEL FLUID MECHANICS IN A FUNGAL PELLET?

Figure 1. Scheme of the averaging process at every position of the acquired images from light microscopy. The process consists on compacting the information of the representative averaging surfaces of the pellets section into points.

Introduction:

Submerged cultures of filamentous fungi can be seen as hierarchical systems with different scale levels [START_REF] Nielsen | Hyphal growth and fragmentation of Penicillium chrysogenum in submerged cultures[END_REF][START_REF] Davidson | Mathematical modelling of mycelia: a question of scale[END_REF]. Fluid mechanics provides the basis for studying the effects of shear stress on growth morphology (as mycelia, clumps, and pellets). It is necessary to show the interconnection between biomass growth, morphology changes (like pellet shaving), mass transport of chemical species, vacuolation, filamentous aggregation size during culture and applied mechanical forces [START_REF] Cui | Modeling and measurements of fungal growth and morphology in submerged fermentations[END_REF]. Fluid mechanics in this systems have scarcely been studied (4), even though they provide valuable information for knowledge building. A promising tool to solve this issue is matematical modeling.

Methodology: An upscaling technique previously reported for porous media was used to

derivate the mathematical model. The mathematical model to be derived is not formulated for a particular fungal species; however, requires information about geometrical features to be solved for a specific system. Light, fluorescence, and electron microscopy techniques were used to produce a representative geometry of hyphae and pellets from submerged cultures of the model fungi L. trichodermophora. Average models were solved in 3D geometry in Comsol Multiphysics ® and validated with direct numerical simulations. Conclusions: L. trichodermophora pellets possess a homogeneous inner core, where a representative elementary volume could be defined, susceptible to a priori modeling with the upscaled mass and momentum balance equations. The derived mathematical model predicted the average intracellular and extracellular velocities in agreement with direct numerical simulations. No noticeable differences in the velocity fields were observed for the conditions studied here.

Upscaled mass and momentum models

Results and discussion

Identification of homogeneous regions inside the pellet

Figure 4 .

 4 Figure 4. x-component of the dimensionless average velocity in the a), b) intracellular and c), d) extracellular phases, considering a), c) fluid flow through dolipores or b), d) obstructructed dolipores. The error % between the upscaled model predictions and the direct numerical simulations was <5% in all studied cases.

Figure 3 .

 3 Figure 3. Fields of the velocity magnitude of the extracellular fluid and velocity magnitude streamlines of the intracellular fluid, in a periodic unit cell representative of a hyphal compartment divided by septa and an open dolipore.

Figure 2 .

 2 Figure 2. Spatial dependence of the volumetric fraction of the extracellular phase (εγ) in the z and x directions (with origin of the axes at the center of the pellets). Samples of pellets from a) 7 days-old pellet, b) 14 days-old pellet and c) 21 days-old pellet of L. trichodermophora cultures. The blue line indicates the average εγ in each pellet, while the shaded blue region encompasses the corresponding standard deviation of the data. The average εγ of the pellets was, in all cases, arround 0.4.