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The dynamics of continuous microbial culture described by cell age distribution and concentration of one substrate

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Abstract

A mathematical model has been considered in which the known equation of McKendrick and Von Foerster for cell age distribution is combined with that for substrate concentration. The dependence of cell division rate on cell age has been taken as a step function. The interrelation between culture parameters describing the substrate consumption and cell division has been found. The shape of cell age distribution as well as the values of substrate and cell concentrations in steady and transient states have been investigated. Stationary regimes at the initial culture state synchronized by ages have been found to be established as damped oscillations and age waves. Under definite conditions the transition from one steady growth regime to another includes sharp single-time age synchronization of the culture.

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Abbreviations

A :

value of the specific rate of cell divisions,W, at τ>T whenW depends on τ as step function Θ (hr−1)

D :

dilution rate (hr−1)

f(S) :

factor expressing the shape of dependence ρ(S) (dimensionless)

G(τ):

initial shape of the distribution functionn, i.e.n(O, τ) (l−1 hr−1)

K s :

saturation constant (g l−1)

K i :

inhibition constant (g l−1)

m c :

dry mass of an individual cell at arbitrary age (g)

m c0 :

dry mass of an individual cell at zero age (g)

n(t, τ) :

distribution function of cell number concentration by cell age τ at timet (l−1hr−1)

N(t) :

total concentration of cells of all ages at timet (l−1)

P(τ):

probability density of cell division at age τ (hr−1)

S(t) :

concentration of substrate in culture broth at time (g l−1)

S f :

concentration of substrate in fed medium (g l−1)

t :

current time (hr)

T(S) :

age interval beginning from τ=0 within which cells prepare themselves for division but do not divide (hr)

T 0 :

constant term in dependenceT(S) (hr)

T 1 :

coefficient in dependenceT(S) originating from substrate limitation (hr g l−1)

T 2 :

coefficient in dependenceT(S) originating from substrate inhibition (hr g−1 l)

W :

specific rate of cell division (hr−1)

Y X/S :

yield of dry cell mass from substrate (dimensionless)

α:

=AT 0, ratio of the cell cycle phase when they cannot divide,T, to the average duration of the phase of cellular division, 1/A, whenT≈T 0 (dimensionless)

ε:

small coefficient at the derivative in dimensionless equation for substrate concentration (dimensionless)

Θ:

step function, equal to 0 when its argument is negative and 1 when the argument is positive (dimensionless)

ν(t, τ):

normalized distribution function of cell number concentration by cell age τ at timet (hr−1)

ρ:

specific rate of substrate consumption (per cell) (g hr−1)

ρ m :

dimensional coefficient in dependence ρ(S) (g hr−1)

τ:

cell age, the time elapsed from the beginning of cell cycle (hr)

τ1 :

width of initial age distribution functionG(τ) when it is taken as a narrow rectangle (case of synchronized culture) (hr)

′:

dimensionless variables

−:

steady-state values

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  1. Institute of Biochemistry and Physiology of Microorganisms, Russian Academy of Sciences, 142292, Pushchino, Moscow Region, Russia

    I. G. Minkevich & A. Yu. Abramychev

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  1. I. G. Minkevich
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  2. A. Yu. Abramychev
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Minkevich, I.G., Abramychev, A.Y. The dynamics of continuous microbial culture described by cell age distribution and concentration of one substrate. Bltn Mathcal Biology 56, 837–862 (1994). https://doi.org/10.1007/BF02458270

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  • DOI: https://doi.org/10.1007/BF02458270

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