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Statistical models of cell populations

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Advances in Biochemical Engineering, Volume 11

Part of the book series: Advances in Biochemical Engineering ((ABE,volume 11))

Abstract

Statistical models for the description of microbial population growth have been reviewed with emphasis on their features that make them useful for applications. Evidence is shown that the integrodifferential equations of population balance are solvable using approximate methods. Simulative techniques have been shown to be useful in dealing with growth situations for which the equations are not easily solved.

The statistical foundation of segregated models has been presented identifying situations, where the deterministic segregated models would be adequate. The mathematical framework required for dealing with small populations in which random behavior becomes important is developed in detail.

An age distribution model is presented, which accounts for the correlation of life spans of sister cells in a population. This model contains the machinery required to incorporate correlated behavior of sister cells in general. It is shown that the future of more realistic segregated models, which can describe growth situations more general than repetitive growth, lies in the development of models similar to the age distribution model mentioned above.

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Abbreviations

A:

Age of a cell randomly selected from the population

a:

Typical value of A

C:

Environmental concentration vector (m-dimensional)

c:

Typical point in environmental concentration space

Ċ:

Rate of consumption of environmental substances

Cs :

Substrate concentration

E:

Expectation

fr :

Product density of rth order

fC :

Probability density for C

fZ/A :

Probability density for Z conditional on a knowledge of A

fZ/S :

Probability density for Z conditional on a knowledge of S

f i1 :

Product density of order 1 for an “i-let”, defined in 4.1

Jv :

Janossy density or Master density

M:

Mass of cell selected at random from the population

\(\bar \dot M\) :

Average mass-specific growth rate

m:

Typical value of M

N:

Total number of cells per unit volume of culture

n:

Number density function

p:

Partitioning function for physiological state

Pv :

Probability distribution for N

\(\bar R\) :

Biochemical reaction rate vector

r:

Partitioning function for cell size

ri :

Number of “i-lets” per unit volume of culture

S:

Size of a cell selected at random from the population

\(\bar \bar S\) :

Average size-specific growth rate

s:

Typical value of S

t:

Time

V:

Variance

Z:

Physiological state vector (n-dimensional)

z:

Typical point in physiological state space

Ż:

Average growth rate of cell

ℭ:

Environmental concentration space

\(d\mathfrak{c}\) :

Infinitesimal volume in ℭ

\(\mathfrak{S}_s\) :

Hypersurface in physiological state space defined by Eq. (17)

\(d\mathfrak{f}\) :

Infinitesimal surface on \(\mathfrak{S}_s\)

\(\mathfrak{B}\) :

Physiological state space

\(d\mathfrak{v}\) :

Infinitesimal volume in \(\mathfrak{B}\)

β:

Stoichiometric matrix for biochemical constituents of the cell

γ:

Stoichiometric matrix for environmental substances

Γ:

Age-specific or size-specific transition probability

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Authors and Affiliations

  1. School of Chemical Engineering, Purdue University, 47907, West Lafayette, IN, USA

    D. Ramkrishna

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© 1979 Springer-Verlag

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Ramkrishna, D. (1979). Statistical models of cell populations. In: Advances in Biochemical Engineering, Volume 11. Advances in Biochemical Engineering, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-08990-X_21

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  • DOI: https://doi.org/10.1007/3-540-08990-X_21

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08990-2

  • Online ISBN: 978-3-540-35678-3

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