Modeling Plant Growth
BioQUEST 2008
Yaffa Grossman
Beloit College
June 15, 2008
Let’s create a rough model of Wisconsin Fast Plant (Brassica rapa) growth by making some estimates and assumptions.
Assume the following constants:
Leaf area on day 1 is 0.5 cm2.
Instantaneous gas exchange rate for all leaves is 5µmol CO2 m-2 s-1 . This value incorporates gross photosynthesis and respiration by the entire plant. How would you measure such a value?
Conversion of leaf area to leaf mass = 20 g m-2 (0.002 g c m-2)
Carbohydrate mass is (12 + 2*1 + 16) g mol-1 (Why?)
Conversion efficiency has two components: the carbon content and the cost of growth respiration
Growth respiration is 0.211 g CH20 g DW-1
Estimated from value for peach fruits from DeJong and Goudriaan 1989
Plant biomass is 42% carbon. Carbohydrate is 40% carbon.
Why? Calculate the carbohydrate value yourself.
Using this information we can calculate the quantity of carbohydrate needed to make plant biomass:
x g CH20 g DW-1
= 0.42 g C g DW-1 / 0.40 g C g CH20-1 + 0.211 g CH20 g DW-1
= 1.261 g CH20 g DW-1
How much carbon is fixed on day 1?
Assume that 60% of this carbon becomes leaf mass. What is the cumulative mass each day? (Assume that the entire plant mass is in leaves on the first day.)
Compare the leaf mass each day to the leaf area that you use for each day. Are the numbers similar in magnitude? Explain.
Procedure:
Create a spread sheet. Put the constants for photosynthetic rate in units of mol CO2 m-2 s-1, conversion coefficient, leaf mass/area conversion, conversion efficiency for carbohydrate to plant dry mass, and the proportion of new dry mass is allocated to leaf area into cells at top of sheet
Label a column “Day” and add day labels for days 1 through 12.
Label a column “Leaf area, cm2.” Assume that the leaf area is 0.5 cm2 on day 1.
Label a column “Leaf area, m2” and enter a formula that converts leaf area from cm2 to m2.
Label a column “Gas exchange, mol per sec.” Enter a formula that calculates the gas exchange rate for day 1. Place dollar signs in front of the cell components (column letter and row number) that represent constants to fix the address so that it does not change when it is copied.
Label a column “Gas exchange, mol day-1” and enter a formula that calculates the gas exchange that occurs in a day.
Label a column “CHO mass, g day-1” and enter a formula that calculates the mass of carbohydrate fixed each day. Assume one mole of CO2 becomes one mole of carbohydrate.
Label a column “New plant mass, g day-1” and enter a formula that converts carbohydrate to additional plant mass. Use the conversion efficiency of 1.261 g CH20 g DW-1.
Label a column “New leaf mass, g day-1” and enter a formula that represents the new leaf mass.
Label a column “New leaf area, m-2 day-1” and enter a formula that represents the new leaf mass.
Label a column “New leaf area, cm-2 day-1” and enter a formula that represents the new leaf mass in cm-2 day-1.
In the cell that represents leaf area on day 2, enter the sum of the leaf area on day 1 and the leaf area produced on day 1.
Copy the formulas for the other columns into day 2 and then copy all formulas into days 3-12.
Calculate cumulative values:
Label a column “Cum plant mass, g.” In the row for day 1, enter the product of initial leaf area and the mass per unit leaf plus the mass added on day 1. In the row for day 2, enter the sum of the cumulative plant mass on the previous day and the mass added on that day. Copy the formula into the rows for days 3-12.
Label a column, “Cum leaf mass, g” and enter the appropriate formula for day 1 and days 2-12.
What shortcuts have we taken? Were they justified?
Data collected by Plant Ecophysiology 2006 (Aaron Berdanier, Melissa Custic, Leah Feeley, Sam Peake, and Kara Sitton) for Wisconsin Fast Plant leaves and stems. The plants were grown under 24 hour illumination.
Day / Mean(Mass) / Std Err(Mass)2 / 0.0014
4 / 0.00298 / 0.000322
6 / 0.00526 / 0.000453
8 / 0.012325 / 0.001983
10 / 0.02392 / 0.001588
12 / 0.0363 / 0.006478
Exploring Exponential Growth
Enter the mean data collected by the Plant Ecophysiology class.
Create a graph for the data. Add a “Trendline” for the data by highlighting the chart and then selecting “Chart, Add Trendline” option. After selecting the “Type” of trendline, you may click on the “Options” tab and then click on the two “Display” boxes.
Fit an exponential trendline. How well does it fit?
If you would like to linearize the data, you may follow apply the techniques introduced in the next topic. Because Excel uses base e rather than base 10 in the trendline analysis, you’ll need to transform the data using the “=LN”function rather than “=Log”
Working with Logarithms
Use Excel to graph the following data as a scatter plot with the points connected by curved lines.
Time / Population 10 / 1
1 / 10
2 / 100
3 / 1000
Make a second copy of the graph by copying and pasting into the same spreadsheet so that you can change the vertical axis to a logarithmic axis. To do this, double click on the vertical axis, select the “Scale” tab, and check the box next to “Logarithmic scale.” Click on OK. You have produced a semi-log plot in which the horizontal axis is linear and the vertical axis is logarithmic with a base of 10. Note the labels on the vertical tick marks.
Describe the graph on the linear and semi-log sets of axes.
Explanation
Population 1 grew exponentially. In such a case, population size may be expressed using the equation
where Popsize is the population size, InitPopSize is the population size at time 0, 10 is base used for this exponential equation, g is the population growth rate, and t is time.
For our example, InitPopSize=1, g=1, and t=time. Note that when you calculate the slope you must use the logarithm of the value on the vertical axis, not the linear value. For example, for y=100, log10(y)=2.
Try entering the formula with these constants into Excel to demonstrate that it gives the values observed for Population 1.
Logarithms may or may not be familiar to you, but exponents probably ARE familiar. Logarithms are just the mathematical inverse of exponentials. So the equation
may be read as the number to which the number 10 must be exponentiated to get 100. Since 102=100,
.
Logarithms have helpful mathematical properties. For example,
.
Hint: Examine this equation for a=10 and b=100 using base 10 to convince yourself that is correct.
You may use Excel to logarithmically-transform the population size using base 10 with the command “=log(VALUE,10).” Transform the values for Population 1 and graph them.
We saw before that the equation fit the data for Population 1.
Since logarithms are related to exponents, “taking the logarithm” of an exponential equation has an interesting result:
The first transition takes advantage of the following logarithmic property:
The second transition takes advantage of the definition of the logarithm: the exponent to which a base is raised to get the number of interest. For this reason Log10(10gt)=gt.
The final equation is an equation for a straight line with a y-intercept of InitPopSize and a slope of g. This is why the data for population 1, which was growing exponentially, lies along a straight line when it is plotted on a logarithmic scale.
You may use Excel to logarithmically-transform the population size using base 10 with the command “=log(VALUE,10).” Transform the values for Population 1 and graph them.
What are the initial population size (size at time 0) and population growth rate for population 1?
Graph the data for population 2 on the linear and logarithmic vertical axes.
Time / Population 20 / 1
1 / 100
2 / 1000
3 / 1000000
Does population 2 appear to grow exponentially? What evidence did you find?
What are the initial population size (size at time 0) and population growth rate for population 2?
Is it likely that populations grow exponentially over long periods of time? Why or why not?
NOTE: Most studies of population growth use the base e, a transcendental number with an approximate value of 2.71828, because this base has useful properties when it is used in continuous equations. For example, the growth rate, r, calculated with base e is equal to the instantaneous rate of increase or interest rate with continuous compounding.
The inverse of exponentiation with base e is the natural logarithm, written ln.
Since
and
where g is the population growth rate using base 10.
Mass Loss by Peas
Data obtained by Anacelia Saenz.
Mean gas exchange rate for 10 peas = 0.9 μmol CO2 min-1
Rates did not differ on days 1-5 (F1,13=1.9, P=0.19)
0.52 g lost from 10 peas in 4 days
Calculate mass loss per day
Compare to gas exchange rate
Useful information:
Carbohydrate (CH2O) is 12/30 carbon by mass
Carbon’s mass is 12 g mol-1