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Two mathematical models for the development of a single microbial pellet

Part II: Simulation of the pellet growth of Penicillium chrysogenum by a fast method equivalent to a morphological detailed description

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Abstract

Starting from the results of a morphologically detailed description of pellet development, a mathematical model is presented which is expected to yield equivalent results with substantially less computing expenditure. The simplification of the original model (part I of the paper) resulted in an about 60–100fold reduction of the demands for computing capacity. This was achieved by averaging the mycelial morphology within radial layers. Quantities such as cell volume density and substrate consumption rates were taken to be constant within a layer. The description by means of partial differential equations was intentionally omitted except for the mass-transfer into the pellet. The results of the layer model show a far-reaching equivalence to the detailed single-hypha-model. Data from image processing investigations and microprobe measurements of oxygen and glucose in Penicillium chrysogenum pellets correspond to the simulation results. The model appears suitable for further process-simulations with larger ensemble of pellets.

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Abbreviations

C i,R :

concentration of component i (i=O, S) (gl−1)

C i,crit :

substrate concentration i critical for lysis (i = O, S)(gl−1)

C i ,stop :

concentration of substance i below which cells are inactivated (gl−1)

CPU:

central processing unit

d h :

cross-sectional diameter of hyphae (μm)

D eff,i :

effective diffusion coefficient of substance i(i= O, S) (m2 h−1)

D i ,max :

maximal molecular diffusion coefficient of substance i in water at 25 °C (i = O, S) (m2 h−1)

D tips :

transport coefficient of tips (m2 h−1)

hgu⋆ :

standard hyphal growth unit at r = 180 μm (μm)

hgu R :

local hyphal growth unit of layer R (μm)

k i :

Monod coefficient for substrate i (i=O, S) (gl−1)

L i :

total length of branch i (i=1, 2, 3) (μm)

L m :

length of a hyphal segment, Eq. (13) (μm)

L R :

total hyphal length in layer R (μm)

l shear :

constant in Eq. (8)

n R :

current number of growing tips in layer R

n /⋆ g :

number of tips (including inactive and already lysed) in layer R

n s :

number of segments (de novo part of the model)

n shear,R :

number of broken tips in layer R

O :

index for oxygen

pO2 :

pressure of dissolved oxygen (%)

Q i :

uptake rate of substrate i (i=O, S) (gl−1 −1)

R :

index of radial layer (R=1, 2, 3,..., R max)

r :

radius (μm)

R max :

index of the outmost layer of the pellet

r tip :

distance from the pellet centre to the tip position (μm)

r thr :

threshold radius (μm)

S :

index for glucose

SHM:

single hypha model (part I of the paper)

t :

time (h)

t end :

time at the end of the simulation (h)

v r :

volume of layer R (μm3)

Y Mi :

observable yield coefficient of biomass on substrate i (gg−1)

Y Xi :

yield coefficient of biomass on substrate i (gg−1

α :

mean tip extension rate (μm h−1)

α ly :

lysis rate (μm h−1)

α :

maximal tip extension rate, single hyphae model (μm h−1)

γ :

angle between segments of the main hypha

ΔL + R :

increment of hyphal length due to growth/lysis (μm)

ΔL − R :

increment of hyphal length due to shear breakup (μm)

λ shear :

shear force parameter

μ m :

maximal specific growth rate of a hypha (h−1)

ρ s :

shear force parameter

ρ x :

cell mass density (g dry weight per 1 wet cells)

ϕ R (C i ):

normalized growth kinetics on substrate i (i=O,S)

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  1. Institut für Technische Chemie der Universität Hannover, Callinstraße 3, 30167, Hannover, Germany

    J. Meyerhoff & K. -H. Bellgardt

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  1. J. Meyerhoff
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  2. K. -H. Bellgardt
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Meyerhoff, J., Bellgardt, K.H. Two mathematical models for the development of a single microbial pellet. Bioprocess Engineering 12, 315–322 (1995). https://doi.org/10.1007/BF00369508

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