Problem 34 - Calculus for Biology and Medicine

Question

Investigate the advantage of dimensionless variables. The time to the most recent common ancestor of a pair of individuals from a randomly mating population depends on the population size. Let \(t\) denote the time, measured in units of generations, to the most recent common ancestor, and let \(T\) be equal to \(N\) generations, where \(N\) is the population size of the randomly mating population. Define \(z=t / T .\) Show that \(z\) is dimensionless and that the value of \(z\) does not change, regardless of whether \(t\) and \(T\) are measured in units of generations or in units of, say, years. (Assume that one generation is equal to \(n\) years.)

Step-by-Step Solution

Answer

The variable \(z = \frac{t}{T}\) is dimensionless because it is a ratio of two quantities with the same units, which results in unit cancellation, making it a pure number.

Step 1: Define the Original Problem

We are given two variables: \(t\), the time to the most recent common ancestor in units of generations, and \(T = N\), where \(N\) is the population size also in units of generations. We need to show that \(z = \frac{t}{T}\) is dimensionless.

Step 2: Express z as Dimensionless

Dimensionless quantities are those which are pure numbers and thus have no associated physical units. \(z\) is defined as \(z = \frac{t}{T}\). Since both \(t\) and \(T\) are expressed in the same units (generations), the division cancels out the units. Thus, \(z\) remains a pure number, ensuring it is dimensionless.

Step 3: Analyze Units in Terms of Years

If we now consider changing the unit from generations to years, knowing that 1 generation equals \(n\) years, we redefine \(t\) in years as \(t_{years} = t \times n\) and \(T_{years} = T \times n\). Computing \(z\) in years: \(z_{years} = \frac{t_{years}}{T_{years}} = \frac{t \times n}{T \times n} = \frac{t}{T}\). Thus, \(z\) maintains the same value, unaffected by the change of units.

Step 4: Conclude on the Advantage of Dimensionless z

The key advantage of dimensionless numbers like \(z\) is that they simplify problems and allow comparisons across different contexts without worrying about unit conversions. This allows the results to remain valid and comparable, regardless of how time is measured.

Key Concepts

Population Genetics

Population genetics is the study of genetic diversity within populations and the processes that can change this diversity over time. It examines how certain forces, like natural selection, genetic drift, mutation, and migration, can influence gene frequencies.

• **Genetic Drift:** This is a random process that can lead to great changes in small populations. Over many generations, genetic drift might lead a population to lose genetic variation, which can have significant effects on survival and reproduction.
• **Natural Selection:** Unlike genetic drift, natural selection involves non-random changes. It favors traits that enhance survival and reproduction. Over generations, beneficial traits become more common.
• **Mutation:** This is the source of new genetic variations. While most mutations might not affect the population significantly, occasional beneficial mutations can introduce favorable traits.
• **Migration:** Gene flow, or the movement of genes between populations, can introduce new genetic variations into a population. It's essential for maintaining genetic diversity, especially in smaller populations.

Understanding population genetics is crucial for predicting how populations will evolve over time, especially in response to environmental changes and human influence.

Most Recent Common Ancestor

The most recent common ancestor (MRCA) of a set of individuals is the last individual from whom all individuals in this set are directly descended. In population genetics, this concept helps scientists study lineage and evolutionary history.

• **Estimating MRCA:** The MRCA can be estimated using both genetic data and mathematical models. By calculating genetic variations, scientists can deduce how long ago two people likely shared a common ancestor.
• **Relevance of Population Size:** The time to MRCA is heavily influenced by population size. In larger populations, there's generally more genetic diversity, which tends to prolong the time to MRCA. Conversely, smaller populations tend to have a more recent MRCA.

Applying these concepts helps researchers chart the human lineage and migration patterns, understanding how we are all interconnected.

Generations vs. Years

The concept of generations vs. years is essential in understanding how time is measured in population genetics. Generations relate more directly to biological processes, whereas years provide a standard chronological framework.

• **Generation Definition:** A generation is the average time between a parent's first offspring and their child’s first offspring. For humans, this is generally between 20 to 30 years.
• **Variable Conversion:** Relating generations to years can involve converting timescales, which is simple using definitions. For instance, knowing one generation equals 25 years helps translate genetic findings into common chronological terms.
• **Implications in Research:** Using generations as a timescale is often more appropriate for genetic studies. It aligns with reproduction patterns and evolutionary changes, presenting a more biologically meaningful measure over strict calendar years.

By understanding generations, researchers can better interpret genetic data and evolutionary phenomena, bridging the gap between genetic lineage and historical timeline.