6.1.1  Cell populations

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A small population of unicellular organisms presented with abundant resources and ample space will increase exponentially (Figure 6.1a). Population doubling time Td (hours or days) is a function of an inherent capacity for cell division and enlargement which is expressed according to environmental conditions, and in Figure 6.1(a) doubling times for these two populations are 1 and 2 d for fast and slow strains respectively.


Figure 6.1 A population of cells unrestricted by space or substrate supply will grow exponentially. In this hypothetical case, a fast-growing strain of a single-celled organism with a doubling time of 1 d (relative growth rate (RGR) of 0.6932 d-1) starts on day 0 with a population of n cells which increases to 120.n by day 7. The slow-growing strain with a doubling time of 2 d (RGR = 0.3466 d-1) takes twice as long to reach that same size. When data for cell numbers are ln transformed, exponential curves (a) become straight lines (b) where slope = RGR.

Exponential curves such as those in Figure 6.1(a) can be expressed as




where N(t) is the number of cells present at time t, N0 is the population at time 0, r determines the rate at which the population grows, and e (or Euler’s number) is a transcendental number where e = 2.7182 and is also the base of natural logarithms. By derivation from Equation 6.1



and accordingly is called relative growth rate with units of 1/time. Doubling time can be shown to be Td=(ln 2)/r.

If a population or an organism has a constant relative growth rate then doubling time is also constant, and that population must be growing at an exponential rate given by Equation 6.1. The ‘fast’ strain in Figure 6.1(a) is doubling every day whereas the ‘slow’ strain doubles every 2 d, thus r is 0.6932 d–1 and 0.3466 d–1, respectively.

If cell growth data in Figure 6.1(a) are converted to natural logarithms (i.e. ln transformed), two straight lines with contrasting slopes will result (Figure 6.1b). This application of ln transformation is a crucial concept in growth analysis, providing a basis for calculation of growth indices discussed later. For strict exponential growth where N(t) is given by Equation 6.1, it follows that



so that a plot of ln N(t) as a function of time t is a straight line whose slope is relative growth rate r.

In practice, r is inferred by assessing cell numbers N1 and N2 on two occasions, t1 and t2 (separated by hours or days depending on doubling time — most commonly days in plant cell cultures), and substituting those values into the expression


which expresses r in terms of population numbers N1 and N2 at times t1 and t2, respectively.

If relative growth rate r is not constant, then growth is not exponential but the concept of relative growth rate is still useful for analysis of growth dynamics in populations or organisms. Equation 6.3 is then used to compute average relative growth rate between times t1 and t2 even though population growth might not follow Equation 6.1 in strict terms. In that case plots analogous to Figure 6.1(b) will not be straight lines.