Statistical models of cell populations

  • D. Ramkrishna
Conference paper
Part of the Advances in Biochemical Engineering book series (ABE, volume 11)


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.


Product Density Sister Cell Population Balance Equation Correlate Behavior Transition Probability Function 




Age of a cell randomly selected from the population


Typical value of A

German Symbols


Environmental concentration vector (m-dimensional)


Typical point in environmental concentration space


Rate of consumption of environmental substances


Substrate concentration




Product density of rth order


Probability density for C


Probability density for Z conditional on a knowledge of A


Probability density for Z conditional on a knowledge of S


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


Janossy density or Master density


Mass of cell selected at random from the population

\(\bar \dot M\)

Average mass-specific growth rate


Typical value of M


Total number of cells per unit volume of culture


Number density function


Partitioning function for physiological state


Probability distribution for N

\(\bar R\)

Biochemical reaction rate vector


Partitioning function for cell size


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


Size of a cell selected at random from the population

\(\bar \bar S\)

Average size-specific growth rate


Typical value of S






Physiological state vector (n-dimensional)


Typical point in physiological state space


Average growth rate of cell

Environmental concentration space


Infinitesimal volume in ℭ


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


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


Physiological state space


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

Greek Symbols


Stoichiometric matrix for biochemical constituents of the cell


Stoichiometric matrix for environmental substances


Age-specific or size-specific transition probability


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Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • D. Ramkrishna
    • 1
  1. 1.School of Chemical EngineeringPurdue UniversityWest LafayetteUSA

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