## Sunday, October 21, 2012

### Plant Interest

Here is a seedling. It has two leaves which collect sunlight for photosynthesis.  Some of the sugar created is used for the plant’s metabolism and some is used to create DNA, protein, cellulose, et cetera to make new cells and hence new leaves.

The plant at time ‘zero’.

The plant at week 1.

The plant at week 2.
The plant at week 3.

The plant at week 4.

 Time (weeks) Leaf area (arbitrary units) 0 2 1 3 2 4 3 5 4 6

The plant is adding one new leaf every week.  As it originally had two leaves, its growth rate can be considered to be 50% per week.
To create a formula we will
·         Use the letter P to represent the original number of leaves.  Therefore P = 2.
·         Use the letter R to represent the growth rate.  Therefore R = 50% =  0.5
·         Use the letter T to represent the number of weeks. Therefore 0 <= T <= 4.
·         Use the letter I to represent the number of new leaves.
·         Use the letter A to represent the total number of leaves.

Our formula for plant growth is I = PRT. Consequently, A = P + I.
 T I = PRT A = P + I 0 I = 2 × 0.5 × 0 = 0 A = 2 + 0 = 2 1 I = 2 × 0.5 × 1 = 1 A = 2 + 1 = 3 2 I = 2 × 0.5 × 2 = 2 A = 2 + 2 = 4 3 I = 2 × 0.5 × 3 = 3 A = 2 + 3 = 5 4 I = 2 × 0.5 × 2 = 2 A = 2 + 2 = 4 n I = 2 × 0.5 × n = n A = 2 +  n

As you can see, the formula matches perfectly with the observed growth.  Some of you may be saying those formulas look suspiciously like the simple interest formulas for the growth of money in a bank account.  Well, yes.  Actually, these are valuable plants which produce an anti-malarial drug.  Consequently, each leaf can be sold for exactly \$1. 00.

End of story?  Not really.  A biologist examines the data. He notices that in week 0 the plant produces 200 units of sugar, i.e. 100 units per leaf.  From the 200 units,  the plant used 150 units for metabolism and 50 units to grow a new leaf.  In succeeding weeks the plant produced much more sugar. However every week the plant only allocated the same 50 units to growth.

 T Sugar produced Sugar used for metabolism Sugar used for growth of a new leaf Percentage of sugar used for growth 0 200 150 50 25% 1 300 250 50 17% 2 400 350 50 13% 3 500 450 50 10% 4 600 550 50 8% n 200 + 100 n 150 + 100n 50

The biologist notes growth is tending towards 0% as n becomes larger.  Does this not contradict the previous statement that the growth rate R remains at a constant 50% every week? No, because we are ‘comparing apples and oranges.’ The 50% growth rate is in comparison to the original size of the plant. The declining growth rate noted by the biologist is recalculated on a week-to-week basis.

The owner of the plant asks the biologist to use DNA technology to increase the plant’s growth.  After months, or possibly years, of failed experiments, the genetically modified plant grows as follows.

 Time (weeks) Leaf area Growth on a week by week basis 0 2 n/a 1 3 (3-2)/2 × 100 = 50% 2 4.5 (4.5-3)/3 × 100 = 50% 3 6.75 (6.75-4.5)/4.5 × 100 = 50% 4 10.125 (10.125 - 6.75)/6.75 × 100 = 50%

Some readers may object to the leaf area being given as 4.5 in week 2.  Does this imply ½ a leaf? A moment’s consideration will show there is no cause for concern. The original plant’s leaves were all 10 cm2 in size. In the genetically modified plant, most of the new leaves are smaller than the original two leaves, but each leaf still contributes some growth. So in some sense, “Yes, ½ a leaf.”

Financially sophisticated readers will see that the biologist has improved the plant’s yield by causing the plant to switch from a linear growth model to an exponential growth model.

The sequence 2, 4, 6, 8, 10,  … exhibits linear growth. The equation of a line is y = mx + c. This equation is of exactly the same form as A = PI + P. We match y with A, m with P, x with I and c also with P.

The sequence 2, 4, 8, 16, 32,  … exhibits exponential growth. The equation of an exponential curve is y = k×bx. This equation is of exactly the same form as A = P(1 + r)n. We match y with A, k with P, (1 + r) with b and x with n. In conclusion, not only is it the case that “time is money” but plant growth is also money!