GROWTH KINETICS

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).

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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 bell-shaped 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 straight-line 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

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

= n₂ - n₁yields average slope over that time interval

t₂ - t₁

Relative Growth Rate (RGR)

RGR = dn _· 1

dt n

= ln n₂ - ln n₁yieldsconstant slope during logarithmic phase

t₂ - t₁

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 area₂ - leaf area₁ = LA₂ -LA₁

interval plant dry weight₂ - plant dry weight₁ W₂ - W₁

; units = cm² g^-1 or cm²/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 W₂ - ln W₁

LA₂ - LA₁t₂ - t₁

W₂ - W₁

=W₂ - W₁ _·ln W₂ - ln W₁_;units = g cm^-2 day^-1 or g/cm²/day

LA₂ - LA₁ t₂ - t₁

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 = cm²_leaf cm^-2_soil or cm²_leaf/cm²_soil

soil area A

Measures the fraction of crop cover.

LAI is near 0 at planting, and is usually 2-3 at full canopy coverage

Crop Growth Rate (CGR)

CGR = NAR · LAI ; units = g cm^-2_soil day^-1 or g/cm²_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 dm² leaf area per day)
Plant Type	Average Over Growing Season	During Main Growth Phase
C4 Grasses	>200	400-800
Herbaceous C3 Plants Grasses Dicots	50-150 50-100	70-200 100-600
Woody Dicots Topical and Sub-Tropical Deciduous Temperate Trees Conifers Ericaceous Shrubs	10-20 10-15 3-10 5-10	30-50 30-100 10-50 15
CAM Plants	2-4	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 6-fold decrease in NAR and tries to compensate by increasing its LAR (i.e. produces about 2-fold more and/or larger leaves), but the RGR still decreases dramatically. At low light intensities, NAR of the shade plant only decreases 3-fold, 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/cm²/ wk	%	cm²/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

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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 plant-plant 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 3-2, Leopold and Kriedemann 1975).

	Tomato		Bean
	300 ppm CO₂	1,000 ppm CO₂	300 ppm CO₂	1,000 ppm CO₂
RGR (mg g^-1 d^-1)	222	254	122	172
NAR (mg dm^-2 d^-1)	71	89	46	80
LAR (dm² g^-1)	3.0	2.8	3.2	2.7
root/shoot ratio	0.19	0.21	0.18	0.25