Question

Why do elephants live longer than mice? Let \(L\) represent the lifespan of an organism. It is known that all mammalian species live, on average, to have about \(1.5 \times 10^{9}\) heartbeats (HB), independent of their size. Assume that an organism will have a total of \(\mathrm{HB}=L \times\) \(\mathrm{HR}=\) constant heartbeats during its lifetime, where HR is its heart rate. What assumptions would you need to make to explain the observation that the lifespans of organisms grow with body mass \(M\) as \(L \propto(\mathrm{HB} / \mathrm{HR}) \propto M^{1 / 4}\) ?

Short answer

Larger animals live longer because their slower heart rate compensates for their larger body mass (L ∝ M^{1/4}).

Step-by-step solution

- Understand the Problem

The problem asks why larger animals like elephants live longer than smaller ones like mice, given the constant total number of heartbeats in a lifetime (1.5 x 10^9 heartbeats). We need to explain this using the relationship between lifespan (L), heart rate (HR), and body mass (M).

- Introduce Given Relationships

Given: 1. Total heartbeats (HB) during an organism's lifetime is constant, approximately 1.5 x 10^9. 2. Total heartbeats can be expressed as HB = L × HR. 3. Lifespan, L, is proportional to body mass, M, raised to the power of 1/4: L ∝ M^{1/4}.

- Relate Heart Rate with Body Mass

From empirical observations, heart rate, HR, is often inversely proportional to the 1/4 power of body mass: HR ∝ 1 / M^{1/4}. This means that larger animals with greater body mass have slower heart rates.

- Substitute the HR Relationship

Using the relation HR ∝ 1 / M^{1/4}, substitute into the equation for HB. Since HB = L × HR is constant:L ∝ M^{1/4} and HR ∝ 1 / M^{1/4}.Therefore, HB = (M^{1/4}) × (1 / M^{1/4}) = constant, which confirms the given observation.

- Draw Conclusion

The assumption that heart rate is inversely proportional to the 1/4 power of body mass and the proportionality of lifespan to body mass to the 1/4 power explains why larger animals like elephants, with slower heart rates, live longer than smaller animals like mice.

Key concepts

heart rate

Heart rate (HR) is the number of heartbeats per unit time, typically measured in beats per minute (bpm). For mammals, an interesting empirical observation shows that heart rate varies inversely with body mass raised to the 1/4 power. This means that smaller animals with smaller body masses have higher heart rates, while larger animals have slower heart rates. For example, a mouse has a much faster heart rate compared to an elephant.

The relationship can be expressed as:

\(\text{HR} \propto \frac{1}{M^{1/4}}\)

This formula suggests that as the body mass (M) increases, the heart rate decreases. This is a key factor in understanding why different animals have different lifespans despite having roughly the same number of total heartbeats over their lifetime.

lifespan proportionality

Lifespan (L) is the duration of time an organism is expected to live. In mammals, lifespan is found to be proportional to the body mass (M) raised to the 1/4 power. This relationship is denoted as:

\(\text{L} \propto M^{1/4}\)

This means that an animal's lifespan increases with its body mass, but not in a direct one-to-one relationship. Instead, the lifespan grows at a slower rate compared to the increase in body mass. For example, an elephant, which is much heavier than a mouse, will live longer, but not proportionally as much longer as the difference in their masses.

Combining this with the total number of heartbeats, since larger animals have slower heart rates, they spread their heartbeats over a longer period, resulting in a longer lifespan.

mammalian biology

Mammalian biology includes the study of various physiological traits that are common within the mammalian class. One of these traits is the intriguing relationship between heart rate, body mass, and lifespan. All mammalian species empirically average about 1.5 × 10^9 heartbeats over their lifetimes, regardless of size.

The concept can be summarized through these points:
* Heart rate is inversely proportional to the 1/4 power of body mass.
* Lifespan is proportional to the 1/4 power of body mass.
* When considering these together, the product of heart rate and lifespan (total heartbeats) remains constant across different species.

This unifying principle helps to explain why smaller mammals with fast heart rates tend to live shorter lives compared to larger mammals with slower heart rates. It is an elegant illustration of how different biological systems maintain balance and proportionality, showcasing the interconnectedness of the mammalian body's functions.