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| Increase - a compensation of decrease |
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Geometric increase is not as common as decrease, but under good conditions life and money show this property. This page deals with a special form of life: Bacterial reproduction or - if the bacterial population is regarded as a whole - bacterial growth. It was already demonstrated that continuous dilution by one vessel unit will cause a decrease of the concentration cStart by a factor e. This will be true for a non-growing bacterial population in the same way. To compensate for this dilution, the population must grow by a factor e: |
| 1. Dilution | 2. Growth |
| cdiluted = cstart * e-1 | cstart = cdiluted * e1 |
| Normally growth is based on time and characterised by a rate and a period. The following examples a) - c) demonstrate that different base variants are possible to specify growth by a rate (x) and a period of 1 hour. In other words: no base is absolute. |
| a) | c = c0 * 2x * 1 | twofold increase per 1 h = doublings / h |
| b) | c = c0 * ex * 1 | "e-fold" increase per 1 h |
| c) | c = c0 * 10x * 1 | tenfold increase per 1 h |
| Within the formulas a) and b), with base 2 and base e respectively, rates with well-known names, given by Monod about 50 years ago, and other symbols than simply x are in use. Students have to accept the given names, symbols and explanations from textbooks: |
| Formula | Rate designation | Symbols |
| c = c0 * 2 ν * t | "growth rate" | ν ρ |
| c = c0 * e μ * t | "specific growth rate" | μ |
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The variant with base e is important for continuous culture. The specific growth rate µ ("e-foldings"/h) compensates for the dilution rate D ("e-sections"/h): c = c0 * e-Dt * eµt or c = c0 * e(µ - D) * t . If the sequence of rates in the combined exponent on the right is used as the sequence of explanation (µ followed by D), the instruction will be suboptimal. The natural connection of e with continuous dilution, characterized by D, was already demonstrated, and µ specifies an equivalent compensation process. An explanation of µ without continuous culture and D ends up in a lame comparison: If one assumes a stepwise doubling of bacteria per hour (or money per year by interest) - which are both rather artificial assumptions - and this stepwise doubling is then made continuous, one may find e for batch growth (or continuous compound interest). But µ cannot really be explained graphic without continuous dilution and this is the dilemma of authors of microbiological textbooks: They have to start with batch culture and not with continuous culture. All comparisons to explain µ for a batch culture are lame and not graphic. |
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