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Two mathematical models for the development of a single microbial pellet

Part I: Detailed morphological model based on the description of individual hyphae

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Abstract

A mathematical model for pellet development of filamentous microorganisms is presented, which simulates in detail location and growth of single hyphal elements. The basic model for growth, septation and branching of discrete hyphae is adopted from Yang et al. [2, 23]. Exact solutions to the intracellular mass-balance equations of a growth-limiting key component is given for two types of either branched or unbranched cellular compartments. Furthermore, the growth model was extended in regard to the external mass-balance equations of limiting substrates (oxygen, glucose) under the assumption that the substrates can enter the denser regions of the pellet only diffusively. Penetration of the substrates into the more porous outer regions of the pellet occurs more easily due to microeddies in the surrounding fluid. Chipping of hyphae from the pellet surface by shear forces was included in the model as well. The application of shear forces leads to a marked smoothing of the simulated pellet surface. The development of pellets from spore germination up to late stages with cell-lysis due to shortage of substrates in the pellet centre can be described. The effects of various model parameters are discussed.

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Abbreviations

A i :

algebraic coefficient (i = 1, 2,..., 6)

B i :

algebraic coefficient (i = 1, 2,..., 6)

C i :

mass-concentration of component i (i = O2, S) (gl−1)

C i,crit :

concentration of substance i critical for lysis (i=O2, S) (gl−1)

C i,stop :

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

C(l i,t) :

intracellular concentration of the key component at site l i and time t (gl−1)

C m :

maximal intracellular concentration of the key component (gl−1)

C X :

Concentration of dry biomass (gl−1)

D :

intracellular diffusion coefficient of the key component (m2 h−1)

D max,i :

maximal molecular diffusion coefficient of substrate i (i = O2, S) (m2 h−1)

D eff,i :

effective diffusion coefficient of component i (i = O2) (m2 h−1)

d h :

cross-sectional diameter of hyphae (m)

k :

production coefficient for the key component (h−1)

K s :

Monod coefficient for glucose (gl−1)

k 0 :

Monod coefficient for oxygen (gl−1)

L c :

total length of a compartment (m)

L i :

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

l i :

position on branch i (i=1, 2, 3)

L m :

maximal length of a segment (m)

m i :

maintenance coefficient of substrate i (h−1)

N m :

maximal number of segments in a compartment

n iR :

number of tips of type i in layer R, i=1, 2

p :

auxiliary variable (see Eq. (7))

P Br :

probability that a hypha is chipped off (%h−1)

pO 2 :

partial pressure of oxygen in the liquid phase (%)

Q :

auxiliary variable (see Eq. (8))

Q i :

uptake rate of substrate i (i = O2, S) (gl−1 h−1)

q :

auxiliary variable (see Eq. (7))

R :

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

r :

radius (m)

r crit :

critical radius, Eq. (15) (m)

r max :

pellet radius (m)

r tip :

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

r thr :

threshold radius (m)

s :

auxiliary variable (see Eq. (7))

S :

index for glucose

t :

time (h)

v R :

volume of layer R (1)

Y Mi :

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

Y Xi :

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

α i :

actual tip expansion rate (m h−1)

α i,m :

actual maximal extension rate of tip i (i=1, 2) (m h−1)

α 1y :

lysis rate (h−1)

α m :

maximal tip extension rate (m h−1)

β :

auxiliary variable in Eq. (2)

γ :

auxiliary variable in Eq. (3)

κ :

auxiliary variable defined in Eq. (4) (m−1)

λ shear :

shear force parameter

μ R :

overall specific growth rate in layer R (h−1)

μ m :

maximal specific hyphal growth rate (h−1)

ρ :

cell volume density (l cell volume per 1)

ρ crit :

critical cell volume density in Eq. (15)

ρ S :

shear force parameter

ρ X :

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

ϕ(C i) :

growth kinetics on substrate i

ψ :

proportional factor in Eq. (34) (l g−1)

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Authors and Affiliations

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

    J. Meyerhoff, V. Tiller & K. -H. Bellgardt

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  1. J. Meyerhoff
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  2. V. Tiller
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  3. K. -H. Bellgardt
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We thank the Deutsche Forschungsgemeinschaft (DFG) for financially supporting parts of this work.

We thank the Deutsche Forschungsgemeinschaft (DFG) for financially supporting parts of this work.

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Meyerhoff, J., Tiller, V. & Bellgardt, K.H. Two mathematical models for the development of a single microbial pellet. Bioprocess Engineering 12, 305–313 (1995). https://doi.org/10.1007/BF00369507

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