Growth Curves

Growth curves for populations of many organisms, including duckweeds, can be divided into different phases (labeled A - D), as shown in the graph.

• Phase A is the lag phase.  This phase occurs after plants are transferred to fresh medium with abundant nutrients.  During this phase, the organism prepares to grow.  Unseen biochemical changes, cell division and differentiation of tissues occur during this time.
• Phase B is the exponential growth phase.  During this phase the plants are growing, producing new fronds and dividing rapidly to take advantage of fresh medium.
• Phase C is the transitional phase.  During this phase growth slows or the death rate increases.  As a result, the initiation of new fronds and the senescence and death of old ones start to come into equilibrium.  For duckweeds this phase typically occurs as nutrients become limiting for growth.
• Phase D is the equilibrium phase.  After this time, birth of new fronds and death of old ones is in equilibrium.  New fronds will only appear as nutrients are released from the death and decay of old fronds.

For natural populations, there may be fluctuations in the availability of nutrients over the course of the growing season.  For example, nutrients may be diluted after heavy rains.  Applications of fertilizers in nearby fields and subsequent runoff may result in periodic bursts of nutrients into the water.
 

Why measure growth curves?

Growth curves must be drawn from a series of growth measurements at different times during the growth curve.  This is much more time-consuming than just measuring a single end-point.  Why go to the trouble?

Using total growth end-point measurements in bioassays is misleading.  There are two reasons for this:

1. If the control culture reaches transitional or equilibrium phase (see above), the assay is invalid.  Given enough time, a slower-growing culture may reach the same final biomass as an initially faster one.  Unless intermediate time points are recorded for a growth curve, the difference in growth rates will be missed.
2. In an end-point assay the "No Observable Effects Concentration" (NOEC) of a toxin will depend strongly on the test duration and the absolute growth rate of the controls.  End-point measurements do not allow for the fact that duckweed growth is nearly exponential, not linear.

Are duckweed growth rates exponential?

• Duckweed growth can be nearly exponential (Phase B above).  Exponential growth is defined as growth where the number of individuals doubles on average in a given time.  If you start with a young duckweed frond, the growth will be very nearly exponential.  However, for exponential growth to occur every individual must continue to reproduce during a given interval of time.
• Duckweed fronds do not exactly satisfy that definition because each frond can only produce a finite number of daughter fronds before it ages and dies.  Actively-growing duckweed cultures often contain one or more of these senescent fronds.  You can recognize them because they turn yellow after they stop growing.
• Early and late daughters of an original frond may not grow equally well, since they may have arisen under different environmental conditions (depletion of nutrients, metabolism of toxins, exposure to other experimental conditions).  This difference does not mean that the early and late fronds differ in their genetic growth potential, since they are all genetic clones.  Nevertheless, later-developing fronds may show altered growth rates in response to environmental conditions.
• In a duckweed growth experiment, only a small number of fronds can be counted.  Therefore, the exponential growth equation is only an estimate of what would occur in a larger population.

Equations for Ideal Exponential Growth

Exponential growth, the number of fronds, N, at a given time t is given by Equation 1, where:  N_o= number of starting fronds and k = exponential growth rate constant.  This can be re-arranged to give Equations 2 and 3.  An estimate of doubling time ( t₂ ) can be calculated  from Equation 4, where ln(2) = 0.693.  A better estimate of k and t₂can  be obtained from a linear regression analysis using Equation 2.  Linear regression analysis is easily done in spreadsheets and many scientific calculators.

Equation 1:    N = N_oe^kt
Equation 2:    k·t = ln N - ln N_o 
Equation 3:    k = ( ln N - ln N_o ) / t
Equation 4:    t₂ = ln 2 / k

Keep in mind that these equations are valid only for ideal exponential growth conditions (Phase B above).  For other phases of growth, different sets of equations must be applied to compensate for lag phase or to allow for senescence and death.
 

Counting Fronds

The most commonly used method of measuring growth of duckweed is to count fronds.  Most duckweed research and phytotoxicity tests depend on frond counts.  However, counting fronds is deceptively easy.  To count the fronds in just a few containers can take considerable time.  Frond counts will be proportional to biomass only if fronds in different treatments share the same average measurements (the same geometry) and same density.  The effects of the different treatments may negate these assumptions.

When counting fronds, it is the accepted procedure to count every visible frond, even the tips of small new fronds that are just beginning to emerge from the pocket of the mother frond.  The "blackboard" drawing shows an example with several fronds in different orientations and stages of growth, and with two plants connected by a stipe (stipule).

A magnifying glass or a stereomicroscope (10x is good) is necessary for good frond counts.  Recall how much care is necessary in counting blood cells in a hemocytometer.  It is all to easy to miss fronds or count them twice.  Placing a square-ruled sheet of paper beneath the culture can help to reduce counting errors.  It is also possible to automate frond counting using video image analysis.

This [ growth curve ] was obtained by counting fronds.
 

Weight, Fresh or Dry

Weight is an obvious measure of plant growth.  However, weighing duckweed plants can be a problem.  Keep in mind that duckweeds are delicate.  Even measuring their fresh weight requires removing the plants from their medium and blotting them.  Thigmotropism, the response of plants to mechanical force and vibration, is a possible result of even gentle handling.  Thigmotropism is known to cause changes in growth rate or the pattern of growth in many plants.  It is unknown if this response occurs in duckweeds, but genes regulating this response have been identified in Arabidopsis.

Fresh or wet weight is measured after blotting the plants very gently with a soft paper towel to remove the free moisture.  Then they are weighed immediately.  Since duckweed lack a cuticle on their lower surface, waiting even a short time will allow drying.  Duckweeds are between 86% and 97%water by weight (Landolt, 1987, page 9), so small variations in drying will have a major impact on results.  However, with care, fresh weights are a useful means of estimating the biomass.

Dry weight is a more acceptable means of measuring growth, but is useful only for an end point.  Keep in mind that duckweed dry weight is just 3% to 14% of fresh weight (see above), so (for laboratory-scale experiments) an accurate milligram balance will be necessary.  According to Landolt (1987, page 69), the dry weight of the largest duckweed fronds (Spirodela) are about 0.6 mg each.

Determine dry weight by blotting the plants and placing them on a pre-weighed and numbered weighing boat. Dry the plants in a hot oven overnight.  Unless rigorous research is being done, the temperature need not be very high.  It is only necessary that the duckweeds be dried to a uniform condition every time.  Since duckweeds have a high surface area and no cuticle on the lower surface, they dry rapidly.  It is generally unnecessary to weigh them more than once to show that the weight is constant.  However, I do place them in a desiccator to cool before weighing them.  A Zip-Loc^TM bag would also work  This method is suited to obtaining weights on very small samples of duckweeds grown in the lab.

To accelerate the process, use a microwave oven and/or a draft of dry air (a hair dryer) - but don't blow the duckweed away!   Also part-way through the process, fork the still-moist plant material from the bottom up to the top.
 

Surface Area

Since duckweeds grow by spreading over the surface of their medium, surface area is a useful means of estimating growth.  As with frond counts, surface area measurements will be proportional to biomass only if fronds in different treatments share the same average measurements (the same geometry) and same density.

Surface area can be estimated from enlarged photographs or tracings, or can be calculated using digital image analysis.  One simple way of estimating area in enlargements and tracings is to cut out the enlarged duckweeds, place them on graph paper and count the squares covered.  Another is to weigh the cutouts.  Following such procedures will give students an appreciation for the ease of digital imaging methods.

Surface area measurements are the only reasonable way to measure duckweed growth on a large scale, as on bayous, ponds and canals.  Airborne false color video analysis has successfully been employed to study the growth of duckweeds in Dutch canals by Synoptics.  For application to environmental management the results were incorporated into a Geographical Information System (GIS).
 

Digital Image Analysis

If a digital video camera or a flat-bed scanner is available, these methods can be used.  Photographs can be scanned off-line for later analysis.  Commercial software is available from many vendors for all-purpose scientific image analysis, but will require programming for this application.  Digital image analysis results (in units of cm² or percentage of surface covered) can be calibrated against any of the traditional methods for measuring duckweed biomass.  Where such a correlation breaks down (for example where growth conditions result in production of smaller fronds), the results (in area/frond) may be interesting in their own right.

LemnaTec, GmbH, a company in Germany, has commercialized measurement of duckweed growth by digital image analysis.
 

References

Landolt, E. and Kandeler, R. (1987) The family of Lemnaceae - a monographic study.  Vol. 2, Phytochemistry, physiology, application, bibliography.  Veroff. Geobot. Inst. ETH, Zurich, 638 pp.
 
 
 
 
 
 
 
 

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Revised:  May 3, 2003