Introduction to Biological Formulae

Mathematics is a vital tool for any biologist. In your OCR AS Level Biology exams, at least 10% of the marks come from mathematical skills. While some complex formulas are given to you in a data sheet, there are several key biological formulae that you are expected to memorize and recall from scratch. This chapter focuses on these essential formulas, particularly those involving rates of reaction, the \(Q_{10}\) coefficient, and physiological calculations.

Don't worry if math isn't your favorite subject! We will break these down step-by-step. Think of these formulas as "recipes"—once you know the ingredients and where they go, the calculation becomes much easier.

Note: For formulas regarding Magnification or Surface Area to Volume Ratio, please refer to the chapter "Magnification, scale and surface area to volume ratio".

1. Calculating Rates

In biology, we often want to know how fast a process is happening. This could be how quickly an enzyme breaks down a substrate, how fast a plant transpires, or how quickly a person breathes.

The Formula

\( \text{rate} = \frac{\text{change in quantity}}{\text{time taken}} \)

How to use it:

  • Change in quantity: This is how much "stuff" was produced or used up (e.g., volume of oxygen produced in \(cm^3\)).
  • Time taken: How long the process lasted (e.g., seconds or minutes).

Example: If an enzyme produces \(20 cm^3\) of product in \(4\) minutes, the rate is \(20 / 4 = 5 cm^3 \text{ min}^{-1}\).

Quick Tip: Always check your units! If the question asks for the rate "per second" but gives you "minutes," you must multiply the minutes by \(60\) before dividing.

2. The Temperature Coefficient (\(Q_{10}\))

The \(Q_{10}\) value tells us how much the rate of a reaction increases when the temperature is raised by \(10^\circ C\). For many biological reactions, the rate roughly doubles for every \(10^\circ C\) rise (meaning \(Q_{10} \approx 2\)).

The Formula

\( Q_{10} = \frac{R_2}{R_1} \)

Where:

  • \(R_2\): The rate of reaction at the higher temperature (e.g., at \(25^\circ C\)).
  • \(R_1\): The rate of reaction at the lower temperature (e.g., at \(15^\circ C\)).

Crucial Rule: The difference between the two temperatures must be exactly \(10^\circ C\).

Step-by-Step Example:
If the rate of an enzyme reaction is \(1.5 \text{ units/s}\) at \(20^\circ C\) and \(3.0 \text{ units/s}\) at \(30^\circ C\):
\( Q_{10} = 3.0 / 1.5 = 2 \)
This means the reaction speed doubled.

3. Heart and Lung Formulae

You will need to recall specific formulas relating to the transport systems in animals (Module 3).

Cardiac Output

This is the total volume of blood pumped by the heart per minute.

\( \text{cardiac output} = \text{heart rate} \times \text{stroke volume} \)

  • Heart Rate: Number of beats per minute (bpm).
  • Stroke Volume: The volume of blood pumped out of the left ventricle in one beat (usually in \(cm^3\)).

Respiratory Quotient (RQ)

This tells us which type of substrate (like carbohydrates or lipids) an organism is using for respiration.

\( RQ = \frac{CO_2 \text{ produced}}{O_2 \text{ consumed}} \)

Did you know? An RQ of \(1.0\) usually suggests the organism is respiring carbohydrates, while an RQ of \(0.7\) suggests lipids.

4. Growth and Biodiversity Formulae

These formulas appear in Module 4 and are essential for analyzing populations and genetics.

Microorganism Population Growth

Bacteria divide by binary fission, meaning the population doubles every generation. To calculate the number of individuals (\(N\)) after a certain number of divisions (\(n\)):

\( N = N_0 \times 2^n \)

  • \(N\): The final number of microorganisms.
  • \(N_0\): The initial number of microorganisms.
  • \(n\): The number of generations (divisions) that have occurred.

Proportion of Polymorphic Gene Loci

This is a measure of genetic biodiversity within a species.

\( \text{proportion of polymorphic gene loci} = \frac{\text{number of polymorphic gene loci}}{\text{total number of loci}} \)

Terminology check: A locus (plural: loci) is simply the position of a gene on a chromosome. A polymorphic locus has more than one allele (version of a gene).

5. Essential General Mathematics

Beyond biological formulas, you must recall standard GCSE-level math applied to biology contexts.

Percentage Change

Used to compare results, such as the change in mass of a potato cylinder in an osmosis experiment.

\( \text{percentage change} = \frac{\text{new value} - \text{original value}}{\text{original value}} \times 100 \)

Common Mistake: Always divide by the original value, not the new one!

Percentage Error

Used to show the uncertainty in your measurements from equipment like pipettes or thermometers.

\( \text{percentage error} = \frac{\text{uncertainty}}{\text{measured value}} \times 100 \)

Quick Review: What to Remember

The "Must-Memorize" List:

  • Rate: \( \text{Change} / \text{Time} \)
  • \(Q_{10}\): \( R_2 / R_1 \)
  • Cardiac Output: \( \text{Heart Rate} \times \text{Stroke Volume} \)
  • RQ: \( CO_2 / O_2 \)
  • Bacterial Growth: \( N_0 \times 2^n \)
  • Genetic Diversity: \( \text{Polymorphic Loci} / \text{Total Loci} \)

What you DON'T need to memorize:
Don't panic about Simpson’s Index of Diversity, Standard Deviation, or Statistical Tests (t-test, Chi-squared). The specification confirms these formulas will be provided in the assessment if you need them. Your job is to know how to use them and how to find the "critical values" in a table!

Summary Takeaway

Biological formulas are just ways of describing living processes using numbers. Always write down the formula before you start a calculation; even if you make a mistake in the math, you might still get a mark for selecting the correct "recipe"!