biology

The Michaelis–Menten curve $v = V_{max}[S]/(K_m + [S])$: the rate rising towards $V_{max}$, ruled across the plot, with $K_m$ marked where the rate is half of it.

The Lineweaver–Burk plot: Michaelis–Menten inverted, so the curve becomes the straight line $1/v = (K_m/V_{max})(1/[S]) + 1/V_{max}$. Across is $1/[S]$ and up is $1/v$. The line is drawn back past the rate axis to meet the concentration axis at $-1/K_m$, and crosses the rate axis at $1/V_{max}$.

A competitive inhibitor, drawn dashed over the uninhibited curve: the inhibitor competes for the active site, so the apparent $K_m$ rises to $K_m(1 + [I]/K_i)$ while $V_{max}$ is unchanged — enough substrate still outcompetes it.

A noncompetitive inhibitor, drawn dashed over the uninhibited curve: the inhibitor binds away from the active site, so the apparent $V_{max}$ falls to $V_{max}/(1 + [I]/K_i)$ while $K_m$ is unchanged — the half-maximum point stays above the same concentration.

The Hill equation $\theta = [L]^n/(K^n + [L]^n)$: the fraction of sites bound, against ligand concentration. $n = 1$ is a hyperbola; above 1 the binding is cooperative and the curve turns sigmoid. Every curve passes through half saturation at $K$, which marked drops to both axes.

Haemoglobin's oxygen dissociation curve: percentage saturation against the partial pressure of oxygen, a Hill curve with $P_{50}$ — the pressure at half saturation — dropped to the pressure axis. The defaults are adult human haemoglobin, in mmHg.

The Bohr shift, drawn dashed over oxygen-dissociation: lower pH or more carbon dioxide moves the curve right by by, so haemoglobin gives up more of its oxygen at the same pressure.

A dose–response curve, the four-parameter logistic $y = b + (t - b)/(1 + (EC_{50}/x)^s)$, with $EC_{50}$ marked halfway between the floor and the ceiling. Drawn on a log dose axis — give the graph x scale: log, where the curve is the symmetric sigmoid it is taught as.

Exponential growth $N = N_0 e^{rt}$, with the starting population marked and the doubling time $\ln 2/r$ dropped to the time axis. r above zero.

Logistic growth $N = K/(1 + ((K - N_0)/N_0)e^{-rt})$: the population levelling off at the carrying capacity $K$, ruled across the plot, with the inflection at $K/2$ — where growth is fastest — dropped to both axes. The inflection comes at $t = \ln((K - N_0)/N_0)/r$, after the start only when $N_0$ is below $K/2$.

The three survivorship curves: survivors from a cohort against age, on a log survivors axis. Type I holds on until old age (large mammals), type II loses the same fraction at every age and is straight on the log axis (many birds), and type III loses most of the cohort young (fish, plants). All three start at the whole cohort and reach one survivor at lifespan. Give the graph y scale: log and a y domain from 1 to the cohort.

The Lotka–Volterra predator–prey system in its phase plane, prey across and predators up: $x' = \alpha x - \beta xy$, $y' = \delta xy - \gamma y$. The field's arrows show the flow, a solution circles the equilibrium from a seed the reader can drag, and the two nullclines are ruled across the plot, meeting at the equilibrium $(\gamma/\delta, \alpha/\beta)$.

Genotype frequencies under Hardy–Weinberg equilibrium against the frequency $p$ of allele $A$: $p^2$ for $AA$, $2pq$ for $Aa$ and $q^2$ for $aa$, with $q = 1 - p$. The heterozygotes peak at a half where $p = q$. marked rules a line at at and marks the three frequencies on it.

A food web: write each species as a vertex whose group is what it eats — producer, herbivore, omnivore, carnivore, decomposer — and each feeding link as an edge from the food to the eater, the way energy flows. The web is laid out in trophic levels running left to right, producers first, and each group takes its colour in the key. A group not in that list is dealt a colour of its own.

A phylogenetic tree: write each taxon and each ancestor as a vertex, the root first, and each branch as an edge from ancestor to descendant. The tree runs left to right with square branches, each name set past its tip. lengths reads each edge's weight as its branch length, so the tree is a phylogram; off, every branch is one long and the tree is a cladogram, which shows only the branching order. Give the ancestors a small size, so the branches meet at a point rather than a disc.

Snowshoe hare and Canada lynx pelts traded to the Hudson's Bay Company, 1900–1920, in thousands — the predator–prey cycle every ecology course shows, the lynx peaking a year or two after the hares. year, hare, lynx.

Mendel's second-generation pea counts (1866), one row per trait: each trait's dominant and recessive forms and how many plants showed each, every trait near three to one. trait, dominant, recessive, dominant-count, recessive-count.