Problem 89 - Calculus for Biology and Medicine

Question

Adapted from Benton and Harper, 1997) In vertebrates, embryos and juveniles have large heads relative to their overall body size. As the animal grows older, proportions change; for instance, the ratio of skull length to body length diminishes. That this is the case not only for living vertebrates, but also for fossil vertebrates, is shown by the following example: Ichthyosaurs are a group of marine reptiles that appeared in the early Triassic and died out well before the end of the Cretaceous. \({ }^{1}\) They were fish shaped and comparable in size to dolphins. In a study of 20 fossil skeletons, the following allometric relationship between skull length \(S\) (measured in \(\mathrm{cm}\) ) and backbone length \(B\) (measured in \(\mathrm{cm}\) ) was found: $$ S=1.162 B^{0.93} $$ (a) Choose suitable transformations of \(S\) and \(B\) so that the resulting relationship is linear. Plot the transformed relationship, and find the slope and the \(y\) -intercept. (b) Explain why the allometric equation confirms that juveniles had relatively large heads. (Hint: Compute the ratio of \(S\) to \(B\) for a number of different values of \(B-\) say \(, 10 \mathrm{~cm}, 100 \mathrm{~cm}, 500 \mathrm{~cm}-\) and compare.)

Step-by-Step Solution

Answer

The allometric equation suggests juveniles had larger skulls as the \(S/B\) ratio decreases with increasing body size \(B\).

Step 1: Understand the Allometric Equation

The given equation is \( S = 1.162 B^{0.93} \). It describes a power-law relationship between skull length \( S \) and backbone length \( B \). This suggests an allometric relationship where the skull grows in proportion to the body but at a different rate.

Step 2: Linearize the Equation

To transform the allometric equation into a linear form, take the logarithm of both sides: \( \log(S) = \log(1.162) + 0.93 \log(B) \). This equation is now of the form \( y = mx + c \) where \( y = \log(S) \), \( x = \log(B) \), \( m = 0.93 \) (slope), and \( c = \log(1.162) \) (y-intercept).

Step 3: Plot the Transformed Relationship

Prepare a plot with \( \log(B) \) on the x-axis and \( \log(S) \) on the y-axis. Plotting these transformed variables gives a straight line with slope 0.93 and y-intercept \( \log(1.162) \). The slope indicates how the skull grows relative to the body as size increases.

Step 4: Verify the Slope and Y-intercept

Calculate \( \log(1.162) \) to determine the y-intercept, which is approximately \( 0.064870 \). Therefore, the linear equation is \( \log(S) = 0.93 \log(B) + 0.064870 \). This confirms the slope as 0.93 and the y-intercept as 0.064870.

Step 5: Compute \(S/B\) Ratios for Different \(B\) Values

Calculate \( S \) using the original equation for \( B = 10 \text{ cm}, 100 \text{ cm}, \text{ and } 500 \text{ cm} \). - For \( B = 10 \), \( S = 1.162 \, (10)^{0.93} \approx 8.626 \). - For \( B = 100 \), \( S = 1.162 \, (100)^{0.93} \approx 85.112 \). - For \( B = 500 \), \( S = 1.162 \, (500)^{0.93} \approx 393.228 \).Now compute the ratios: - \(\frac{S}{B}\) for \(B=10\) is \(0.8626\),- \(\frac{S}{B}\) for \(B=100\) is \(0.85112\),- \(\frac{S}{B}\) for \(B=500\) is \(0.786456\).

Step 6: Analyze the \(S/B\) Ratio

The \( S/B \) ratio decreases as \( B \) increases. This confirms the hypothesis: juvenile ichthyosaurs have proportionally larger skulls compared to their body size, as the head grows at a slower pace than the body as they mature.

Key Concepts

Logarithmic Transformations

Logarithmic transformations are essential tools in data analysis, especially when dealing with non-linear relationships like allometric equations. They simplify the equations by converting complex multiplicative relationships into easier-to-handle additive relationships.
In the context of ichthyosaurs, we have an equation for skull length in relation to backbone length: \( S = 1.162 B^{0.93} \). The goal is to linearize this relationship for easier interpretation and analysis. To do this, we apply a logarithmic transformation to both sides of the equation.

• By taking the logarithm of both sides, we derive: \( \log(S) = \log(1.162) + 0.93 \log(B) \). This conversion transforms our original power-law into a linear format resembling the equation of a line, \( y = mx + c \).
• Logs convert exponential growth patterns into straight lines, making trends more apparent and allowing for the calculation of meaningful growth rates with slopes and intercepts.

With this transformation, the relationships become much more straightforward to interpret, allowing for easy plotting and further analysis.

Linearization of Equations

Linearizing equations converts non-linear relationships into linear ones through mathematical transformations. This process involves modifying variables so that they behave predictably over a certain range.Let's consider the linearization within our problem involving ichthyosaurs. Once we've applied a logarithmic transformation, we're left with \( \log(S) = 0.93 \log(B) + 0.064870 \). This new equation is linear, where:

• \( m = 0.93 \) represents the slope, indicating that skull length grows at a slightly slower rate than backbone length.
• \( c = 0.064870 \) is the y-intercept, reflecting the baseline log-transformed value of \( S \) when \( B \) equals one.

Linearization is significant because it allows us to construct a line graph with clear understanding. It aids in visualizing data by plotting \( \log(B) \) versus \( \log(S) \) and confirming the growth pattern with a straight line. The more predictable nature of linear equations makes them favorable for performing further statistical analysis and predictions.

Juvenile Vertebrate Proportions

The proportions of juvenile vertebrates, such as ichthyosaurs, are important for understanding their growth patterns and evolutionary biology. When vertebrates are young, they exhibit different body proportions compared to their adult forms, like having larger heads relative to their body size.In our study of ichthyosaurs, we observe that the skull-to-body length ratios decrease as the vertebrates grow, highlighting their juvenile dimensions. As demonstrated with calculated ratios for different backbone lengths:

• Smaller ichthyosaurs (e.g., at \( B = 10 \text{ cm} \)) exhibit a larger \( S/B \) ratio, around 0.86, illustrating relatively large skulls compared to their body.
• As they mature to \( B = 500 \text{ cm} \), the ratio decreases to approximately 0.79, indicating that the body grows at a quicker pace than the skull.
  

This changing ratio suggests that early developmental stages prioritize essential sensory and cognitive development, possibly requiring larger initial cranial spaces. As vertebrates grow, their bodies catch up to maintain balance in their form and function. This insight is important for paleontologists and evolutionary biologists when determining growth traits and habits in extinct species.