GROWTH KINETICS
AND
QUANTITATIVE MEASUREMENTS OF GROWTH
REFERENCES
Growth Kinetics
and
Quantitative Measurements of Growth
Larcher, W. 1980. Physiological Plant Ecology,
Leopold, A.C. and P.E. Kriedemann. 1975. Plant Growth and
Development. McGrawHill, Inc., NY.
Wareing, P.F. and I.D.J. Philips. 1981. Growth and Differentiation in Plants,
GROWTH KINETICS
Growth  an irreversible increase in size, mass or number.
Many growth phenomena in nature exhibit a logarithmic or exponential increase. The size, mass or number increases by a constant, similar to simple compound interest. The principal (current size, mass or number) times the interest rate (growth rate) yields the interest (growth increase for that day). The interest is added to the principal, to yield a new principal. The new principal times the interest rate yields and even higher interest for the next day, which again is added back to the principal. So growth occurs at a compounded rate (logarithmic or exponential growth).
Absolute Growth Rate (AGR)
If you plot growth (size, mass or number) versus time, a constantly increasing growth curve is obtained. If you calculate the slope between any two times, you get the absolute growth rate, which is the change in actual growth over time. You get a different slope, hence different AGR for each pair of times chosen to calculate the slope. (Fig. 2.23A, Wareing and Philips 1981)
Relative Growth Rate (RGR)
If you plot the logarithm of growth (size, mass or number) versus time, a linear line is obtained. If you calculate the slope of the line, you get the relative growth rate, which is the change in relative growth over time. Since the line is linear, you get the same RGR, regardless of which time interval chosen to calculate the slope. (Fig. 2.23A, Wareing and Philips 1981).
GROWTH KINETICS con't
Sigmoidal Growth Curve
Exponential growth can never be sustained indefinitely. Eventually, substrates are depleted, the population exceeds the area available, tissues or individuals begin to die, etc., which decreases the growth rate. Growth may still increase, but at a reduced rate (ex. if crowding causes shading), it may reach a steady state (everything is in equilibrium, for example in a population), or growth may begin to decrease (ex. due to death or senescence of individuals or plant parts). If you plot long term growth versus time you get the classical sigmoidal growth curve. If you plot the logarithm of the sigmoidal growth curve, you get a linear line during the exponential phase, after which the curve decreases over time. (Fig. 2.24, Wareing and Philips 1981)
Changes In Growth Rates Over Time
If you calculate the absolute growth rate (AGR) over increments of time, then plot AGR versus the time interval, you get a bellshaped curve, i.e. the AGR changes constantly with time. If you calculate the relative growth rate (RGR) over increments of time, then plot RGR versus the time interval, you get a straightline region during the logarithmic phase followed by a decreasing RGR. The RGR is constant during the logarithmic phase. (Fig. 2.27, Wareing and Philips 1981).
MATHEMATICAL MODELS OF GROWTH
Exponential Model
n = n_{o}e^{kt}
Linear Model – Used During Logarithmic or
Exponential Phase
ln n = ln n_{o} +
(slope) (time)
where n = number, size (height, leaf area), or mass (dry weight, fresh weight) at any time > 0.
n_{o} = number, size (height, leaf area,), or mass (dry weight, fresh weight) at time = 0.
slope = rate of growth
Or more commonly expressed as the slope
equation
y = a + bx
y = intercept + (slope) (x)
Absolute Growth Rate (AGR)
AGR = dn
dt
= n_{2}  n_{1}_{ }yields average
slope over that time interval
t_{2}  t_{1}
Relative Growth Rate (RGR)
RGR = dn _{·} 1
dt n
= ln n_{2}
 ln n_{1}_{ }yields_{ }constant slope
during logarithmic phase
t_{2}  t_{1}
QUANTITATIVE MEASUREMENTS OF GROWTH
Leaf
Area Ratio (LAR)
a) over
life of crop LAR = final leaf area = LA
final
plant dry weight W
b) over
any time LAR = leaf area_{2}  leaf
area_{1} = LA_{2}
LA_{1}
interval plant dry weight_{2}  plant dry weight_{1} W_{2}  W_{1}
; units = cm^{2} g^{1} or cm^{2}/g
LAR is an indication of the efficiency of a
given leaf area to produce a given plant size.
Net
Assimilation Rate (NAR)
NAR
= RGR = 1 _{· }RGR
LAR LAR
= 1 _{· } ln W_{2}
 ln W_{1}
LA_{2}
 LA_{1}_{ }t_{2}
 t_{1}
W_{2}  W_{1}
=_{ }W_{2}  W_{1} _{·
}ln W_{2}  ln W_{1}_{
; }units = g cm^{2} day^{1} or g/cm^{2}/day
LA_{2}
 LA_{1} t_{2}
 t_{1}
NAR measures the accumulation of plant dry weight per unit leaf area
per unit time.
It is a measure of efficiency of production.
Leaf Area
Index (LAI)
LAI = leaf
area = LA ; units
= cm^{2}_{leaf} cm^{2}_{soil} or cm^{2}_{leaf}/cm^{2}_{soil}
soil area A
Measures the fraction of crop cover.
LAI is near 0 at planting, and is usually 23 at full canopy coverage
Crop Growth Rate (CGR)
CGR = NAR · LAI
; units = g cm^{2}_{soil}
day^{1} or g/cm^{2}_{soil}/day
CGR measures the efficiency of production of a
total field of plants over a given soil area.
APPLICATION OF QUANTITATIVE
MEASUREMENTS OF GROWTH
Efficiency of Different Species of
Plants
The following table gives the net assimilation rates (NAR) of various
species. The higher the NAR the more
efficient the species, which usually translates into higher growth rates. (from Table 3.10, Larcher 1980)

Net Assimilation Rate (mg dry matter per dm^{2} leaf area per day) 

Plant Type 
Average Over Growing Season 
During Growth Phase 
C4 Grasses 
>200 
400800 
Herbaceous
C3 Plants 
50150 50100 
70200 100600 
Woody
Dicots 
1020 1015 310 510 
3050 30100 1050 15 

24 
10 
Efficiency of Sun versus Shade
Plants
The following table gives the net assimilation rates (NAR), leaf area
ratio (LAR), and relative growth rate (RGR) of shade versus sun plants at both
high and low light intensities. (from
Table 3.1, Leopold and Kreidmann1975).
Note: At low light intensities,
the sun plant has 6fold decrease in NAR and tries to compensate by increasing
its LAR (i.e. produces about 2fold more and/or larger leaves), but the RGR
still decreases dramatically. At low
light intensities, NAR of the shade plant only decreases 3fold, and increases
its LAR 2.4 fold, both of which help maintain a higher RGR; in other words the
shade plants have adapted themselves to the lower light intensity.

NAR 
LAR 
RGR 

% Daylight 
mg/cm^{2}/ wk 
% 
cm^{2}/g 
g/g wk 
% 
Sun Plant  Sunflower 

100% 24% 12% 
8.0 2.9 1.3 
100 36 17 
82 140 170 
0.66 0.42 0.23 
100 64 35 
Shade Plant  Impatiens 

100% 24% 12% 
6.1 3.3 2.0 
100 54 33 
132 239 315 
0.80 0.78 0.63 
100 98 79 
APPLICATION OF QUANTITATIVE
MEASUREMENTS OF GROWTH  con't
Effect of Leaf Area Index (LAI) on
Net Assimilation Rate (NAR) and Crop Growth Rate (CGR)
Note that as the LAI increases (due to greater canopy coverage of soil), the NAR (productivity of each plant) decreases (probably due to increased plantplant shading), but the CGR (productivity of the entire crop over a given area of soil) increases. Thus, the best LAI is somewhere around 4.
(from Fig. 3.64, Larcher 1980)
Use of Quantitative Growth Measurements to Explain
Other Growth Phenomena
Increasing ambient carbon dioxide increases photosynthesis, which in turn increases growth. In tomato and bean, increasing carbon dioxide increases both total plant growth, as measured by increased RGR, and the efficiency of growth, as measured by increased NAR. This increased growth efficiency allows the plant to have a smaller shoot system (decreased LAR), which is the source, while still enhancing the size of the root system (see increased root/shoot ratio), which is a sink (from Table 32, Leopold and Kriedemann 1975).

Tomato 
Bean 


300 ppm CO_{2} 
1,000 ppm CO_{2} 
300 ppm CO_{2} 
1,000 ppm CO_{2} 
RGR (mg g^{1} d^{1}) 
222 
254 
122 
172 
NAR (mg dm^{2} d^{1}) 
71 
89 
46 
80 
LAR (dm^{2} g^{1}) 
3.0 
2.8 
3.2 
2.7 
root/shoot ratio 
0.19 
0.21 
0.18 
0.25 