Introduction to Math Topics in Biology
Introduction:
The purpose of this exercise is
to examine different types of biological (or other relevant scientific) data and
decide whether the relationships are linear, exponential, or something else.
o
Differentiate between linear and exponential models based on qualitative
shape of curve.
o
Differentiate between the mathematical expression of linear and
exponential models.
o
Understand the consequences of exponential growth.
problems:
1.
Animals vary in their ability to regulate body temperature.
Below are examples of body temperatures for two different animals. Are
these relationships linear, exponential, or something else?
Body temperature vs. ambient (environmental) temperature
|
ambient
temp. (°C) |
Snake body
temp. (°C) |
|
ambient
temp. (°C) |
Bobcat body
temp. (°C) |
|
3 |
2 |
|
5 |
37 |
|
10 |
10 |
|
10 |
38 |
|
15 |
14 |
|
20 |
39 |
|
20 |
19 |
|
30 |
39 |
|
29 |
29 |
|
40 |
39 |
|
33 |
33 |
|
|
|
|
38 |
37 |
|
|
|
2.
Radioactive decay is used to date fossils and thus is an important tool
in some fields of biology. The
following is simulated data for a hypothetical isotope with a half-life of one
day. Actual half-lives may be in
seconds, minutes, or any time unit up to billions of years.
In this example, suppose you start with a 1.0 kg sample of this element
with a half-life of one day. One
day later you will have half of the original sample (the other half will
disintegrate in the first day). Complete
the following table for days 2-8. Is this relationship linear, exponential, or something else?
Radioactive decay
|
Time (days) |
Mass (g) |
|
0 |
1000 |
|
1 |
500 |
|
2 |
|
|
3 |
|
|
4 |
|
|
5 |
|
|
6 |
|
|
7 |
|
|
8 |
|
(from Physical
Science by Bill W. Tillery, p. 335)
3.
Does plant growth, in the following example, reflect a linear,
exponential, or other?
Growth
of a sunflower stem
|
Age
(days) |
Observed
Height (cm) |
|
Age
(days) |
Observed
Height (cm) |
|
17.9 |
|
49 |
205.5 |
|
|
14 |
34.4 |
|
56 |
228.3 |
|
21 |
67.8 |
|
63 |
247.1 |
|
28 |
98.1 |
|
70 |
250.5 |
|
35 |
131 |
|
77 |
253.8 |
|
42 |
169 |
|
84 |
254.5 |
Data
from On Growth and Form (pages 259-261), by D’Arcy Wentworth Thompson,
the compete revised edition, Dover, 1992
4.
As humans age we lose the ability to focus objects close to our eyes.
This is measured as an increase in near-point accommodation. Is this
relationship linear, exponential, or something else?
Age vs. Near-point accommodation
|
Age
(yr) |
Near
point (cm) |
|
10 |
7.5 |
|
20 |
9.0 |
|
30 |
11.5 |
|
40 |
17.2 |
|
50 |
52.5 |
|
60 |
83.3 |
(from Tortora A&P Lab Manual, 4th edition)
5.
Cardiac output (CO) is the amount of blood pumped by each ventricle
(lower heart chamber) in one minute. Cardiac output (in ml/min) is equal to
Stroke Volume (SV, in ml/beat) times Heart Rate (HR, in beats/min or bpm).
(Stroke volume is the amount of blood pumped out of each in a single
heart beat). CO = SV x HR.
Calculate the CO for the HR and SV given in the table below.
Is the relationship between HR and CO linear, exponential, or something
else?
|
CO (ml/min) |
HR (bpm) |
SV (ml/beat) |
|
|
75 |
80 |
|
|
100 |
80 |
|
|
60 |
80 |
|
|
40 |
80 |
|
|
120 |
80 |
6.
In fluid dynamics (which also applies to blood flow) flow is proportional
to pressure and inversely proportional to resistance, BF = BP/PR. Where BF = Blood Flow, PR = Blood Pressure, and PR =
Peripheral Resistance. Peripheral
Resistance is the resistance of the arterial system.
Vascular resistance can be increased by constricting arteries.
Is the relationship between blood flow and blood pressure linear,
exponential, or something else? How
about the relationship between blood flow and peripheral resistance?
7.
Are the relationships between maternal age and Down’s syndrome, etc.
linear, exponential, or other?
Estimates of Rates per thousand live births
|
Maternal
Age (yr) |
Down
Syndrome |
XXY |
XYY |
|
<15 |
1.0 |
0.4 |
0.5 |
|
15 |
1.0 |
0.4 |
0.5 |
|
16 |
0.9 |
0.4 |
0.5 |
|
17 |
0.8 |
0.4 |
0.5 |
|
18 |
0.7 |
0.4 |
0.5 |
|
19 |
0.6 |
0.4 |
0.5 |
|
20 |
0.5-0.7 |
0.4 |
0.5 |
|
21 |
0.5-0.7 |
0.4 |
0.5 |
|
22 |
0.6-0.8 |
0.4 |
0.5 |
|
23 |
0.6-0.8 |
0.4 |
0.5 |
|
24 |
0.7-0.9 |
0.4 |
0.5 |
|
25 |
0.7-0.9 |
0.4 |
0.5 |
|
26 |
0.7-1.0 |
0.4 |
0.5 |
|
27 |
0.8-1.0 |
0.4 |
0.5 |
|
28 |
0.8-1.1 |
0.4 |
0.5 |
|
29 |
0.8-1.2 |
0.5 |
0.5 |
|
30 |
0.9-1.2 |
0.5 |
0.5 |
|
31 |
0.9-1.3 |
0.5 |
0.5 |
|
32 |
1.1-1.5 |
0.6 |
0.5 |
|
33 |
1.4-1.9 |
0.7 |
0.5 |
|
34 |
1.9-2.4 |
0.7 |
0.5 |
|
35 |
2.5-3.9 |
0.9 |
0.5 |
|
36 |
3.2-5.0 |
1.0 |
0.5 |
|
37 |
4.1-6.4 |
1.1 |
0.5 |
|
38 |
5.2-8.1 |
1.3 |
0.5 |
|
39 |
6.6-10.5 |
1.5 |
0.5 |
|
40 |
8.5-13.7 |
1.8 |
0.5 |
|
41 |
10.8-17.9 |
2.2 |
0.5 |
|
42 |
13.8-23.4 |
2.7 |
0.5 |
|
43 |
17.6-30.6 |
3.3 |
0.5 |
|
44 |
22.5-40.0 |
4.1 |
0.5 |
|
45 |
28.7-52.3 |
5.1 |
0.5 |
|
46 |
36.6-68.3 |
6.4 |
0.5 |
|
47 |
46.6-89.3 |
8.2 |
0.5 |
|
48 |
59.5-116.8 |
10.6 |
0.5 |
|
49 |
75.8-152.7 |
13.8 |
0.5 |
(from Ob & Gyn. pocket reference, 1984)
8.
According to Boyle’s Law (the Gas Law) pressure and volume of gases are
related by the following equation: PV=nRT.
In physiological systems we will assume that nRT remains constant (or nRT
= constant), so that PV = constant. Is
this relationship between pressure and volume linear, exponential, or something
else?