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The following original model chooses piecewise constant oxygen consumption rates in a closed, well-mixed $2\mathrm{mL}$ suspension. It assumes no gas exchange, negligible addition volume, a corrected sensor and sufficient oxygen throughout. The curves are invented teaching examples, not measured traces, fitted data or a reconstruction of Figure 6.

Start at $c(0)=220\mu \mathrm{mol}{\mathrm{L}}^{-1}$. On successive one-minute intervals, choose positive slope magnitudes $5,25,5,25,6,32,1\mu \mathrm{mol}{\mathrm{L}}^{-1}\mathrm{mi}{\mathrm{n}}^{-1}$. Integration gives the endpoint concentrations $220,215,190,185,160,154,122,121\mu \mathrm{mol}{\mathrm{L}}^{-1}$. This is just the recurrence $c_{i+1}=c_i-r_i\Delta t$, with $\Delta t=1\min$; the step plot shows $r_i=-dc/dt$. The trace remains above zero by construction.

During the second ADP interval, the total consumption rate isWith a stipulated mitochondrial protein mass of $0.20\mathrm{mg}$, that is $250\mathrm{nmol}\mathrm{mi}{\mathrm{n}}^{-1}\mathrm{m}{\mathrm{g}}^{-1}$. Using a denominator twice as large would halve this reported rate without changing the raw oxygen trace. Comparisons therefore need both loading and the chosen denominator.

For a second bookkeeping check, the modeled post-antimycin A slope of $1\mu \mathrm{mol}{\mathrm{L}}^{-1}\mathrm{mi}{\mathrm{n}}^{-1}$ corresponds to $2\mathrm{nmol}\mathrm{mi}{\mathrm{n}}^{-1}$ background. If that background is additive and unchanged in earlier windows, subtraction gives $48$, $10$ and $62\mathrm{nmol}\mathrm{mi}{\mathrm{n}}^{-1}$ for excess ADP, oligomycin and FCCP, respectively. The oligomycin-sensitive difference is $48-10=38\mathrm{nmol}\mathrm{mi}{\mathrm{n}}^{-1}$. It is a difference in oxygen consumption under the chosen interventions, not an ATP production rate. Converting it into one requires additional coupling assumptions or an independent ATP measurement; the FCCP interval shows why slope alone cannot supply that conversion.

Across the whole seven-minute example, consumption is $V(c(0)-c(7))=198\mathrm{nmol}$, also the sum of the seven interval consumptions. This equality checks the rate-to-amount accounting independently of how the intervals are named. Actual records require attention to calibration, mixing transients, background stability and decreasing oxygen availability; a line that steepens or flattens alone does not identify the cause.

The molar solubility $s$ is the amount of substance of a specified solid dissolved per volume of saturated chemical solution, under stated conditions. It counts dissolved formula units, not the sum of all dissolved ion concentrations. For AB with no initial A or B and no competing chemical reactions, one-to-one stoichiometry gives $c_{\mathrm{A}}=c_{\mathrm{B}}=s$. For an AB${}_2$ solid, the corresponding concentrations are $s$ and $2s$.^[ref]

Molecular diffusion is the component of material transport caused by the random thermal motion of molecules, measured relative to bulk motion of their surroundings. A diffusion coefficient is a property of a particular substance in a particular medium under specified conditions, not a universal speed of spreading. Its dependence on the solute, solvent and temperature is precisely what makes a spreading experiment informative. A random walk idealizes this motion; the Brownian motion description provides a continuous approximation over sufficiently long times. ^[ref]

A motivating experiment: watch dye enter a gel. Samprovalaki, Robbins and Fryer studied rhodamine 6G and methylene blue entering agar gel from two side reservoirs. A covered slab, $10\times 5\times 1\mathrm{mm}$, sat on a controlled thermal stage; the dye was brought to the same temperature. They photographed the gel every ten minutes, averaged image intensity across a strip, and fitted the changing concentration to a diffusion equation model. The cover limited evaporation and stopped dye travelling over the gel surface. Runs at $30$, $50$ and $70{}^{\circ }\mathrm{C}$ were repeated in triplicate. The measured spatial colour change therefore supplied a quantitative transport test, rather than merely showing that dye eventually spreads. Changing lighting also changes image intensity, so the optical response had to be checked. Importantly for interpretation, their constant-diffusion coefficient fit was imperfect and the gel coefficient is an effective transport coefficient, not automatically the value in free water. ^[ref]

A finite-pulse version to test width growth. The following is a proposed teaching variant, not that paper's apparatus or a report of measurements. Make a long, narrow, transparent gel channel of uniform cross-section. Join a thin dyed gel band between two undyed lengths of the same gel, with good contact and no air gaps; the dye should initially fill the cross-section so the dominant concentration variation is along the channel coordinate $x$. After assembly, keep the covered channel horizontal at uniform temperature, away from vibration. The dye is a finite initial pulse: no reservoir supplies more dye. Choose a dye whose image response can be calibrated and whose interaction with the gel is small enough for an approximately constant diffusion coefficient to be plausible.

Photograph the whole pulse at recorded times with fixed lighting, focus and exposure, including a length scale. Use uniformly dyed standards of the same gel thickness and a blank gel to convert image intensity to concentration $C(x,t)$; do not assume raw brightness is proportional to concentration. Average across the channel width and examine several parallel strips to check the one-dimensional approximation. Measure the initial profile instead of treating the band as infinitely thin. For a finite pulse with negligible end effects, define its spatial integral $Q$, centre $\bar{x}$ and width $\sigma$ byHere ${\sigma }^2$ is the variance of the normalized profile $C/Q$, and $\sigma$ is its standard deviation. If $C$ is measured in $\mathrm{mol}{\mathrm{m}}^{-3}$, then $Q$ has units $\mathrm{mol}{\mathrm{m}}^{-2}$; multiplying by the channel cross-section gives the total amount of substance. For the unbounded one-dimensional diffusion equation with constant $D$, conserved $Q$, finite variance and sufficiently decaying tails,Thus plot width squared against time: its slope determines the diffusion coefficient, while its intercept includes the finite starting width. This moment law holds even for a non-Gaussian distribution initially; it follows by applying integration by parts twice to the diffusion equation. ^[ref]

For the particularly readable case of an initially Gaussian distribution centred at zero, uniform advection at speed $u$ and constant diffusion coefficient $D$ give the one-dimensional concentration profileIts centre moves as $\bar{x}=ut$ and its variance grows as ${\sigma }_0^2+2Dt$. Our illustrative choice ${\sigma }_0=0.50\mathrm{mm}$ and $D=0.20\mathrm{m}{\mathrm{m}}^2{\mathrm{h}}^{-1}$ gives widths $0.50$, $0.81$ and $1.36\mathrm{mm}$ at zero, one and four hours. These are calculated values, not a measured dye coefficient. Once $2Dt$ dominates the initial variance, width grows as $\sqrt{2Dt}$: four times the elapsed time gives approximately twice the width. A rectangular starting band will not immediately have a Gaussian distribution, although the same width-squared law still applies under the stated assumptions.

Controls and limits of the interpretation. In the proposed experiment, check that $Q$ stays constant and that the centre remains stationary while the width increases. Include a blank and a uniformly dyed gel photographed over the same period: an apparent signal change there exposes lighting drift, dye fading or evaporation. Repeat with a lower starting dye concentration and a second initial width; agreement of the slopes is a check on constant $D$ and on preparation effects. Keep the pulse far from the channel ends and record its faint tails: truncation and background errors can strongly bias a variance. Poor contact, dye binding, chemical reactions or transport in a liquid film can invalidate the simple model. A gel suppresses bulk liquid motion, but its inferred diffusivity need not equal that in the pure solvent. A stationary centre alone does not exclude symmetric flows, and linear width-squared growth alone does not prove a molecular mechanism: turbulence can also produce an effective diffusive description. These are reasons to combine the profile test with the controlled setup.

Molecularity counts the reacting entities in one elementary reaction: one for a unimolecular step, two for a bimolecular step. An entity can be a molecule, ion, or chemical atom. Solvent entities involved only through solvation are excluded. Molecularity describes a microscopic event; order of reaction describes the concentration dependence of a rate law. Assigning molecularity to a net multi-step chemical equation is therefore inappropriate.^[ref]

The Monod equation is an empirical saturating law for the specific growth rate of a culture limited by one nutrient. Write $s\ge 0$ for its residual nutrient concentration, ${\mu }_{\max }>0$ for the limiting specific growth rate, and $K_s>0$ for the nutrient concentration at half that rate:Here $\mu$ and ${\mu }_{\max }$ have units of inverse time, whereas $K_s$ has the same units as $s$. The notation $K(C)$ in the title denotes the same growth function, with $C=s$, $K_{\max }={\mu }_{\max }$ and $K_0=K_s$. For $s\ll K_s$, $\mu (s)\approx ({\mu }_{\max }/K_s)s$; for $s\gg K_s$, $\mu (s)$ approaches ${\mu }_{\max }$. Its derivative ${\mu }^{\prime }(s)={\mu }_{\max }{K}_s/(K_s+s{)}^2$ is positive: more nutrient increases the specific growth rate, with progressively smaller increments.

For example, take ${\mu }_{\max }=1{\mathrm{h}}^{-1}$ and $K_s=1\mathrm{g}{\mathrm{L}}^{-1}$. Nutrient concentrations of $0$, $1$ and $9\mathrm{g}{\mathrm{L}}^{-1}$ give specific growth rates $0$, $0.5$ and $0.9{\mathrm{h}}^{-1}$. In a chemostat with dilution rate $D=0.5{\mathrm{h}}^{-1}$, the intersection $\mu (s)=D$ selects the residual nutrient concentration $s=1\mathrm{g}{\mathrm{L}}^{-1}$, provided the feed can supply enough nutrient for a positive biomass concentration.

The Monod equation is useful because two measurable parameters describe both nutrient scarcity and saturation. It assumes the other conditions affecting growth remain controlled and does not represent inhibition at high nutrient concentration. Although its formula resembles the Michaelis-Menten equation, a whole culture's specific growth rate combines many processes; the resemblance alone does not derive it from Michaelis-Menten kinetics or identify $K_s$ with a Michaelis constant.

In his 1949 review, Monod presents the hyperbolic rate law as an empirical fit in equation (2), pp. 383-384, and discusses the difficulty of identifying a single growth-controlling reaction. His reported division rates use doublings per unit time; the natural-exponential specific growth rate used here is that division rate multiplied by $\ln 2$. ^[ref].

A morphogen is a signal whose spatial concentration helps specify different biological cell fates or behaviors during biological development. A morphogen gradient provides a positional cue: biological cells at different distances from a source experience different local concentrations and can follow different developmental pathways. A spatial concentration variation alone does not establish that a substance is a morphogen; its graded developmental effects matter. See Alberts et al., Molecular Biology of the Cell, 4th edition (2002): morphogens as graded developmental signals.

The beautiful idea is that one distributed signal can help organize several regions of a developing body. A continuous morphogen gradient can supply the input for distinct domains of gene expression, rather than requiring a separate signal for every position. A morphogen response threshold is a useful idealization of that interpretation: high, intermediate and low concentrations are assigned different outcomes. The morphogen reaction-diffusion equation describes one way to generate a morphogen gradient; the equation is a model of transport and reaction, while the morphogen is the signal being transported.

Detection and interpretation also depend on the receiving biological cell. For an extracellular morphogen, receptors and signal transduction connect the local concentration to a response; receptor binding occupancy is only one step in that connection. The same morphogen gradient can be interpreted differently in different tissues because their developmental context differs. Thus a morphogen response threshold summarizes a particular response in a particular context, rather than being a universal property of the signal. See Scott F. Gilbert, Developmental Biology, 6th edition (2000): positional information and context-dependent interpretation.

An important experimental example came in 1988. Wolfgang Driever and Christiane Nüsslein-Volhard used antibodies to examine the bicoid protein in early fruit-fly embryos. They reported a concentration profile approximately described by an exponential function, highest at the front, with the protein concentrated in cell nuclei. Their abstract proposed diffusion from localized messenger RNA and dispersed degradation as a likely origin of the morphogen gradient, rather than proving a unique transport mechanism. ^[ref]

In a companion study, Driever and Nüsslein-Volhard genetically changed the amount and distribution of bicoid messenger RNA, measured the resulting protein morphogen gradient, and compared developmental maps. Increased local bicoid protein levels shifted prospective anterior structures toward the rear; decreased levels shifted them toward the front. This linked a measured concentration profile to positional effects in biological development. The toy calculations below illustrate that logic with arbitrary numbers, not measured bicoid parameters. ^[ref]