Key Formulas & Equations
Core Concepts
Cell Theory
All living organisms are composed of cells; the cell is the basic unit of structure and function; all cells arise from pre-existing cells. Prokaryotes (bacteria) lack a nucleus; eukaryotes (plants, animals, fungi) have membrane-bound organelles.
DNA & Central Dogma
DNA stores genetic information in sequences of A, T, G, C bases. The central dogma: DNA → (transcription) → mRNA → (translation) → Protein. Genes are segments of DNA that code for specific traits or functional RNA.
Natural Selection
Mechanism of evolution proposed by Darwin. Individuals with traits better suited to the environment survive and reproduce more successfully, passing those traits to offspring. Over time, allele frequencies shift, leading to adaptation and speciation.
Homeostasis
The maintenance of stable internal conditions (temperature, pH, glucose levels, water balance) despite external changes. Achieved through negative feedback loops: a deviation from set point triggers responses that reverse the change.
ATP & Energy Currency
Adenosine triphosphate (ATP) is the primary energy currency of the cell. Energy is stored in the bonds between phosphate groups. Hydrolysis of the terminal phosphate releases energy for cellular work: muscle contraction, active transport, biosynthesis.
Ecological Pyramids
Models showing energy flow through trophic levels. Only ~10% of energy transfers between levels (10% rule). Producers (plants) form the base; primary consumers (herbivores); secondary/tertiary consumers (carnivores); decomposers recycle nutrients.
Worked Examples
Example 1: Hardy-Weinberg Equilibrium
Problem: In a population, 16% show a recessive trait (aa). Find the frequency of heterozygous carriers.
Step 1: q² = 0.16 → q = √0.16 = 0.4
Step 2: p = 1 - q = 1 - 0.4 = 0.6
Step 3: Heterozygote frequency: 2pq = 2(0.6)(0.4) = 0.48
Answer: 48% of the population are heterozygous carriers (Aa).
Example 2: Monohybrid Cross
Problem: Cross two heterozygous tall plants (Tt × Tt). What fraction of offspring will be short (tt)?
Step 1: Set up Punnett square:
T t
T | TT | Tt
t | Tt | tt
Step 2: Genotypic ratio: 1 TT : 2 Tt : 1 tt
Step 3: Phenotypic ratio: 3 tall : 1 short
Answer: 1/4 (25%) of offspring will be short (tt).
Example 3: Enzyme Kinetics
Problem: An enzyme has Vmax = 100 μmol/min and Km = 10 mM. What is the reaction velocity at [S] = 10 mM?
Step 1: Use Michaelis-Menten: v = Vmax[S]/(Km + [S])
Step 2: v = (100)(10)/(10 + 10) = 1000/20
Step 3: v = 50 μmol/min
Answer: At [S] = Km, velocity is exactly half of Vmax = 50 μmol/min.
Example 4: Population Growth
Problem: A bacterial population starts with 100 cells and doubles every 20 minutes. How many cells after 2 hours?
Step 1: Number of generations: 120 min / 20 min = 6 generations
Step 2: N = N₀ × 2ⁿ = 100 × 2⁶ = 100 × 64
Step 3: N = 6,400 cells
Answer: There are 6,400 cells after 2 hours.
Practice Problems
Problem 1
If a cell has 8 chromosomes at the start of meiosis, how many chromosomes will each gamete have?
Answer: 4 (haploid, n = 4 after meiosis I and II)
Problem 2
Calculate the surface area to volume ratio of a spherical cell with radius 0.01 mm.
Answer: SA/V = 3/r = 3/0.01 = 300 mm⁻¹
Problem 3
A gene has alleles A (dominant) and a. If two parents are Aa × aa, what fraction of offspring show the dominant phenotype?
Answer: Punnett square: 2 Aa (dominant) and 2 aa (recessive) → 50%
Problem 4
In a food chain with 10,000 kcal at the producer level, how much energy reaches the tertiary consumer?
Answer: 10,000 → 1,000 (primary) → 100 (secondary) → 10 kcal (tertiary)
Problem 5
What is the net ATP yield from glycolysis (per glucose molecule)?
Answer: 2 ATP (4 produced - 2 consumed in the energy investment phase)
Problem 6
If blood pH drops from 7.4 to 7.1, by what factor does [H⁺] increase?
Answer: ΔpH = 0.3, factor = 10^0.3 ≈ 2 times increase