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Equations for the Age Structure of Growing Populations

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Executive Summary

The following is an analysis of “Equations for the Age Structure of Growing Populations,” by George H. Weiss. This paper models the age structure of a growing cell population taking into account physiological age as well as chronological age that is more commonly studied in literature. The author develops a model for differential equations that can be used to find the population density of a given cell colony. Then, approximations of the fundamental equation are suggested.

Context

Cell colonies are often studied in order to answer a variety of biological questions ranging from human health to species survivability. However, it is often difficult to obtain all of the information necessary to characterize these colonies and all of the cells in them. Given data about individual cells within a colony, it is possible to determine characteristics of the entire population. Conversely, given precise measurements of the population overall, it is possible to determine characteristics of the individual cells. With the current advances in technology, it is possible to obtain incredibly precise data on cell concentrations through time, as well as other information that would help characterize cells within the colony. [1]


Within a cell colony there is a significant amount of variability. This stems from different states that the cells are in, as well as the different ages of the cells. The age of a cell is the time from the instant that a cell acquires all of the characteristics of its given state. Some biochemical reactions, such as those relating to DNA replication and enzyme synthesis, relate directly to the cell cycle, and thus are affected by the age of the cell. Other reactions, such as those that create ATP and structural materials, depend more on the state of the cell than the age. [1]

It can be assumed that cells of a similar age share similar intrinsic properties, such as size, relative to cells of different ages.[2] However, the state of the cell also has a great effect on cell properties. Thus the true age of a cell can be split into two components the chronological age and the physiological age of the cell. The physiological age describes the state of the cell, and there are a variety of factors that are used in empirical studies to account for changes in that aspect of the cell characterization. In the context of cells, volume, mass, and number of chromosome faults have been used as such indicators. [3]


Cells can change states, just as they mature in age. Cells can enter a stage through either change from a different group, immigration or birth. On the other hand, cells leave stages by emigration, growth to a different state, or death. All of these factors play a role in the accretion and loss at each stage.

History [2]

The rate of change of a size of a group of cells is equal to the amount of cells entering the group minus the number of cells leaving the group. If there are m stages, for Index.jpg, then the relationship between the total accretion to each stage Ai.jpg, and the total loss from each stage Bi.jpg, should satisfy Materialbalance.jpg

There is a multitude of models that are based off of the von Foerster model that accounts for the distribution of an age-structured population over time. Then

Natda.jpg

describes the number of cells at time t that have an age between a and Ada.jpg. Then at a time dt after t, the population can be modeled as:

Firsteq.jpg

where Death.jpg corresponds to the rate at which organisms die at time t, at the given conditions. This gets von Foerster’s equation:

Pde1.jpg

And integrating gets:

Pde2.jpg

where Birth.jpg is the generalized birth rate, or the number of cells that are age 0 added to the population per unit time.

Figure 1:Population with respect to time and age


Figure 1 depicts a sample population with respect to time and age distribution, given a certain initial population distribution and birth rate.

Mathematical Model [2]

The before-mentioned models of age structured populations take into account only the chronological age of organisms, and do not consider the environmental impact on the variability of organisms within each stage. Adding in a new parameter corresponding to physiological age, α, would help ease this issue. Weiss aims to model the effects of such factors on the structure of the population.

Weiss’s modifications to the von Foerster model are in the addition of the physiological age so that

Doubleint.jpg

corresponds to the total number of organisms between the chronological ages of a_1 and a_2 and the physiological ages of α_1 and α_2. Then the number density can be represented as follows:

Int1.jpg

The rate of physiological growth in a time interval dt is as follows, representing the rate of the transitions, Transfer.jpg: Psi.jpg Then the transition rate at time t from stage α is:

Phi.jpg

Then

Int2.jpg


The first term corresponds to the organisms still remaining in the given stage and the second term corresponds to the organisms advancing to the given stage form the stages before, and the last term refers to death. Then expanding gets:

Int3.jpg

At Initialcond.jpg as well, so for a given set of parameters, the reproduction rate Reproduction.jpg would give the number of newborn cells at time t to be:

Initialcondn.jpg

So the final equation model is:

Int4.jpg

Analytic solution [3]

Weiss notes that approximate solutions to the partial differential equation above are much simpler than attempting to find a solution similar to the von Foerster equation.

One of his approximations is the Fokker-Planck approximation that assumes the changes in physiological age are relatively small. Then the nth moment of the increase in physiological age in the time from t to t+dt

Mu.jpg

Given the assumptions:


Assumptions.jpg

Then

Int5.jpg

Results [3]

Weiss describes a scenario where the indicator, using which physiological age is determined, is the rate of accumulation of genetic defects are accumulated at time t. Then the following equations show under the presence of r defects:

Accum.jpg

where Omega.jpg is the rate per cell at which defects are accumulated and as before Lambda.jpg corresponds to the death rate at the given conditions, and Rho.jpg is the birth rate of newborns at the given conditions. The boundary condition is:

No.jpg

Then the following assumptions are made:

1. The rate at which defects come about is constant, Mu1.jpg

2. The rate at which cells die is a linear equation with respect to the number of defects r, Lambdacalc.jpg where C and D are constants

3. The cells themselves do not reproduce

Then the model is:

Modeldif.jpg

Given Tran.jpg, the following holds:

Dv.jpg

Given an initial condition, the solution to this first order partial differential equation is:

V.jpg

As in the von Foerster model the assumption that at birth, the physiological age is 0, so there are no defects, the following can be said

Naro.jpg

where Initialn.jpg is the initial number of cells, Diracdelta.jpg is the Dirac delta function and Delta.jpg is the Kronecker delta. So, the initial condition for v is

V2.jpg

Then Superfinalmodel.jpg

Conclusions

The original chronological age-distribution model of population distributions of cell colonies has been modified to account for physiological age. Individual cell growth and development now plays a key role in the contribution to the configuration of the population. There is now a developed equation that gives an exact relationship between physiological age, chronological age, and the population distribution. This relationship would be very beneficial in assessing individual mortality and stage change rate.



References

  1. 1.0 1.1 Bell, George I., and Ernest C. Anderson. "Cell Growth and Division: I. A Mathematical Model with Applications to Cell Volume Distributions in Mammalian Suspension Cultures." Biophysical Journal 7.4 (1967): 329-351.
  2. 2.0 2.1 2.2 Trucco, E. "Mathematical models for cellular systems the von Foerster equation. Part I." The Bulletin of Mathematical Biophysics 27.3 (1965): 285-304.
  3. 3.0 3.1 3.2 Weiss, George H. "Equations for the age structure of growing populations." The Bulletin of Mathematical Biophysics 30.3 (1968): 427-435.