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Mathematical models can be derived that describe the manner in which the number of cells in a population increases.

Consider as an example the following hypothetical situation: Suppose that we begin with a culture medium containing a single cell that grows for some period of time and then divides to produce two daughter cells; in turn, these two cells grow and after an identical period of time, they divide to produce four cells, and so on.

In such a situation, the numbers of cells present in the population would increase exponentially in the following way: 1, 2, 4, 8, 16, 32, 64. . . (Fig. 2-1a).

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In other words, the population of cells would double in number with each generation. Consequently, after any specific length of time (say, from t_{1 }to t_{2}), the number of cells in the population would be given by the equation.

N_{2} – N_{1 }x 2^{g} …(2-1)

where N_{1} is the original number of cells present at time t_{1}, N_{2} is the number of cells present at t_{2}, and g is the number of generations that have occurred during the time interval t_{2}-t_{1}. Figures 2-1a and 2-1b depict graphically the exponential nature of the increase in population size during unrestrained growth.

The situation described above is ideal and does not occur in nature (except perhaps for a limited time following the fertilization of an egg cell), but it can be approached artificially under conditions of synchronous cell growth.

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If a typical expanding population of cells is examined at any instant in time, some cells would be observed to be dividing, others would just have completed division, still others would be preparing to divide, and so on. Divisions of all cells present in the population would not occur during precisely the same interval of time.

**Exponential Growth:**** **

If a large number of cells are cultured together in what is called a “random” culture, the individual cells will be found in various stages of their growth- division cycle or cell cycle. At any moment, the rate at which the number of cells in the culture increases is directly proportional to the number of cells present at that time.

This, of course, presumes a steady state in which the needed nutrients are always available in adequate supply and in which cellular waste products excreted into the cells’ environment do not interfere with the maintenance of normal growth and division. The growth of such a random culture is described by the differential equation

dN/dt = kN …(2-2)

Where N is the number of cells present at time t, dN/dt is the change in cell number with time, and k is a growth constant that is specific for the population. This equation may be solved by integration and yields the expression

2.3 log_{10} (N_{2}/N_{1}) = k (t_{2}-t_{1}) …(2-3)

If we let N_{1} equal the number of cells present in the population at time t_{1} and N_{2} equal the number of cells present at time t_{2}, then equation 2-2 is solved by integration as follows.

Thus, In N_{2} – In N_{1} = k (t_{2} –t_{1}) or

In (N_{2}/N_{1}) = k (t_{2}-t_{1})

By converting to the more familiar base 10, the last equation takes the form of equation 2-3. Equation 2-3 indicates that the growth of the population (i.e., the rate at which the number of cells in the population increases) is exponential.

**Doubling Time****:**

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Although the number of cells in a population increases exponentially with time, different types of cell populations (i.e., different species of microorganisms or cells from different tissues) grow at different rates. Even populations of the same type of cell may grow at different exponential rates if the temperature, nutrients, or other growth conditions vary.

Differences in growth rates are reflected by differences in the value of the growth constant, k, in equation 2-3. A convenient value that expresses the specific rate of growth of a population of cells under a specified set of conditions is the doubling time.

The doubling time is defined as the time required for the number of cells in the population to double during exponential growth. Many cell biologists use the term generation time interchangeably with doubling time. Strictly speaking, however, the generation time is the interval of time between any point in one cell cycle (see later) and the same point in the next cell cycle.

Both the growth constant and the doubling time are specific to a particular cell culture. Although the growth rate (i.e., kN) increases as the population of cells gets larger, the doubling time (and, of course, the value of k) remains the same. An equation for the doubling time may be derived as follows.

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After a time interval equal to the-doubling time has elapsed, the ratio N_{2}/N_{1} is equal to 2; therefore, from equation 2-3

2.3log 2 = kT …(2-4)

where T is the doubling time, t_{2}-t_{1}. Hence,

0.693 = kT …(2-5)

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and

T=0.693/A; …(2-6)

The actual value of k or T may easily be determined when experimental data are used to make a semi-logarithmic plot of cell number versus time (Fig. 2-2).

**Sample Problem:**

Suppose that at time t_{1} the number of cells in a population (i.e., N_{1}) is 62,400 and at time t_{2}, 18.5 hours later, there are 473,000 cells. What is the doubling time for this population of cells? From equation 2-3,

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2.3 log (473,000/62,400) = k(18.5)

2.3 log 7.58= 18.5k

k = 0.1094

The dimensions of k in this instance are hr^{-1}; that is, the number of cells in the population increases by 10.94% per hour. Now from equation 2-6,

T=0.693/0.1094 hr^{-1} = 6.33 hr

Therefore, during exponential growth, the number of cells in the population doubles every 6.33 hours.

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**The Lag Phase of Growth:**** **

Typically, when cells are placed in a nutrient medium that favors their growth and proliferation, exponential growth of the population does not begin immediately. Instead, there is a short interval of time in which there is little or no increase in the number of cells in the population. This time interval preceding exponential growth is known as the lag phase (Fig. 2-3).

The length of the lag phase is quite variable, even when comparing different cultures of the same type of cell. A number of factors are believed to influence the length of the lag phase of the population growth curve.

Experiments with bacteria and other microorganisms have shown that variations in the concentrations of certain constituents of the growth medium, such as carbon dioxide, and certain cations, such as H^{+}(i.e., pH), markedly influence the length of the lag phase.

Therefore, the chemical composition of the nutrient medium influences the time interval that precedes the onset of exponential population growth. The cells that are used to “seed” a new culture are acquired from a previous culture that was at some particular stage of its growth curve. The cells used to start a new culture are referred to as the inoculum.

The stage of the parent culture used to provide the inoculum influences the length of the lag phase. For example, the lag phase of cultures of the bacterium Aerobacter aerogenes is longer when the inoculum is drawn from a previous culture that was in early exponential growth and shorter when drawn from a culture that was in late exponential growth. For the protozoan Paramecium caudatum, little or no lag period is observed when the inoculum consists of cells that had been growing exponentially; however, when the inoculum consists of cells from the stationary phase (see below), a lag period is observed.

Generally, the greater the number of cells in the inoculum, the shorter is the lag period. It has been suggested that this is due to the more rapid accumulation in the nutrient medium of a cellular secretory substance that is required during exponential growth and that reaches a “threshold” concentration earlier when there is a large number of initial cells.

**The Stationary Phase****:**

If cells are growing in a medium whose size and contents are initially fixed, then it is clear that growth cannot continue indefinitely. Nutrients in the medium will eventually be depleted, and potentially harmful metabolic waste products excreted into the medium by the cells will accumulate in high concentration.

Consequently, the cell population eventually reaches some limiting size, and following this the number of cells in the population no longer increases. It should be noted that the attainment of a maximum population size does not necessarily mean that cells are no longer growing and dividing, but rather that any additional cells produced by division are balanced by the death and disruption of other cells. The interval of time in which the number of cells in the population ceases to increase and remains fairly constant is known as the quiescent or stationary phase (Fig. 2-3).

**The Death or Declining Phase****:**

Unless the cells of stationary phase cultures are transferred to fresh medium (or fresh medium is added to the culture), stationary cultures eventually enter a death or declining phase. In this phase, the number of cells lost by death and disruption exceeds the number being produced by cell division. As a result, the number of cells in the culture rapidly diminishes (Fig. 2-3).