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For observed pairings between two labels $A,B$, let $N_{\mathrm{same}}=N_{AA}+N_{BB}$ and $N_{\mathrm{cross}}=N_{AB}+N_{BA}$, where the first label denotes the female's group and the second the male's. A mating assortment contrast isIt lies between $-1$ and $1$. Positive $I$ describes an excess of same-label pairings over cross-label pairings; whether that is evidence of assortative mating depends on the null mathematical model and the assay. The four counts form a contingency table.

Here is an original scope check. Suppose independently paired females and males have probabilities $p_f,p_m$ of carrying label $A$. ThenWith $p_f=p_m=1/2$, the expected value is zero. With $p_f=p_m=4/5$, it is $9/25=0.36$, despite pairing being independent of label. This counterexample concerns unequal pool composition; it does not replace the group introductions and contingency table tests in the dietary divergence and mating isolation in Drosophila pseudoobscura experiment. Even with balanced pools, finite observed samples fluctuate: a positive $I$ alone is not a hypothesis test, much less a universal species boundary.

Negative feedback closes a causal loop whose net effect opposes an initiating change. In signal transduction, activation followed by a downstream inhibition of the earlier activating stage is one example of negative feedback. Loop sign alone does not establish asymptotic stability or complete signaling adaptation; see the worked signaling feedback example.

A useful biological example is chemotaxis feedback regulation: an Escherichia coli biological cell adjusts its response to a sustained attractant instead of remaining at its immediate response. Increased CheA activity promotes CheB phosphorylation; CheB reverses receptor methylation, opposing their activation. After attractant addition, reduced activity instead favors receptor methylation by CheR, restoring activity. This is a return from receptor activity to the machinery regulating that activity. The response need not return exactly at every dose. ^[ref]

A motivating measurement is the chemotaxis FRET assay. CheY and CheZ carrying yellow and cyan fluorescent tags were observed in immobilized Escherichia coli biological cells on a coverslip in a flow chamber. Adding and removing attractant changed the yellow-to-cyan emission ratio, reporting the association of phosphorylated CheY with CheZ. This measures a molecular readout upstream of motor behavior. Comparisons with biological cells lacking CheR, CheB, or both tested the dependence on receptor modification; these perturbations also change starting activity and sensitivity, so they are not pure changes of one feedback gain. ^[ref]

The later chemotaxis feedback regulation experiment used only the Tar receptor and nonmetabolizable methylaspartate, supplied and removed by flow. Long recordings separated slow recovery from incomplete signaling adaptation; correction for bleaching addressed drift in the optical readout. Activity recovered incompletely after sufficiently large additions, with detectable imprecision already at $10\mu \mathrm{M}$ in that preparation. Restricting the receptor type and preventing consumption of the stimulus help interpret the perturbation, but limit transfer to other receptors and conditions. The readout infers CheA activity through CheY-CheZ association, rather than directly measuring every molecular step. ^[ref]

Negative feedback can also overshoot or sustain oscillations if its return arrives too late. The worked signaling feedback example gives the contrasting instantaneous case; the delayed negative feedback model below isolates the effect of timing. Loop sign describes causation, while asymptotic stability describes what happens to displacements over time.

A partial derivative measures the local response of a function to one input while its other inputs stay fixed. For a function $f$ on an open subset of ${\mathbb{R}}^2$, its partial derivatives at $(a,b)$ are the limitswhen the respective limits exist. Each is an ordinary derivative of a one-variable slice: $x\mapsto f(x,b)$ or $y\mapsto f(a,y)$. This is why the other variable is treated as a constant during partial differentiation.^[ref]

For an original example, take the polynomial functionHolding $y$ fixed gives $f_x=2x+y$, while holding $x$ fixed gives $f_y=x+4y$. At $(1,1)$, the height is $f=4$ and the two partial derivatives are $f_x=3$ and $f_y=5$. The values answer different experiments on the same mathematical model: change $x$ alone, or change $y$ alone.

The two exact slices through that point areDividing either height change by $t\ne 0$ gives $3+t$ or $5+2t$; taking the limit $t\to 0$ recovers the two slopes. At $t=0.1$, the finite height changes are $0.31$ and $0.52$, although their linear approximations are $0.3$ and $0.5$. A derivative measures the limiting response, rather than an exact finite change.

For a function of $n$ variables, ${\partial }_{j}f(a)$ varies only its $j$th coordinate: it is the same difference quotient along $e_j$, the $j$th standard basis vector. This gives a practical sensitivity analysis: one can compare how a mathematical model responds to changes in its individual inputs, provided the chosen system of units and the quantities held fixed are stated. Comparing the numerical sizes of two partial derivatives in different units does not by itself identify a physically preferable input.

Existence of the partial derivatives at one point is weaker than differentiability. The example under partial derivatives do not imply continuity shows how both coordinate tests can miss a bad diagonal approach. When the first partial derivatives are continuous in a neighborhood, they do give differentiability there, allowing the total differential to combine all small input changes.

Choose the following original phenomenological mathematical model solely to calculate a paired-pulse readout. It is not experimental data, a fit to the teaching experiment, or a simulation of the Hodgkin-Huxley model. The conditioning action potential defines time zero. Test pulses have fixed width $0.2\mathrm{ms}$; $I$ denotes their inward electric current amplitude. Impose an absolute neuronal refractory period $t_{\mathrm{abs}}=2\mathrm{ms}$. No finite test amplitude generates an event for $0<\Delta <t_{\mathrm{abs}}$. After that hold, stipulate a required amplitudewith $I_{\infty }=0.2\mathrm{nA}$, dimensionless $A=2$ and recovery time constant ${\tau }_r=3\mathrm{ms}$. Emit a test event exactly when $I\ge I_{\mathrm{req}}(\Delta )$, including equality. For this ideal event readout, assign events to pulse onset and assume the conditioning event is successfully evoked. There is no calculated membrane potential, spike waveform, response latency or stochastic success probability.

The exponential decay represents a combined recovery deficit; it is neither the Hodgkin-Huxley model's $h$ nor a measured potassium electrical conductance. One such curve cannot separately identify sodium recovery and persistent potassium electrical conductance. A constant pulse-width amplitude criterion also does not establish an equivalent rule for other pulse widths or voltage trajectories.

At the hold boundary, $I_{\mathrm{req}}(2\mathrm{ms})=0.2(1+2)=0.6\mathrm{nA}$. At $\Delta =4\mathrm{ms}$,at $\Delta =6\mathrm{ms}$,Thus a $0.35\mathrm{nA}$ test fails at $1\mathrm{ms}$ because of the imposed hold, fails at $4\mathrm{ms}$ because recovery is insufficient for that amplitude, and succeeds at $6\mathrm{ms}$. At $4\mathrm{ms}$, a $0.5\mathrm{nA}$ pulse would instead succeed: failure of the weaker pulse was relative refractoriness.

For $I>I_{\infty }$, solve $I\ge I_{\mathrm{req}}(\Delta )$ using the logarithm. The earliest allowed delay isFor $I=0.35\mathrm{nA}$ this is $2+3\ln (8/3)\approx 4.942488\mathrm{ms}$; for $I=0.5\mathrm{nA}$ it is $2+3\ln (4/3)\approx 2.863046\mathrm{ms}$. If $I\ge 0.6\mathrm{nA}$, the imposed $2\mathrm{ms}$ hold is the only delay restriction. If $I\le I_{\infty }$, no finite delay passes the rule: the required amplitude stays strictly above its asymptotic baseline. A finite experimental “fully recovered” time therefore also needs a tolerance and a readout definition.

Physics investigates matter, motion, and interactions by building mathematical models and testing their consequences against observations. A physical law supplies a relation; a model specifies the objects, conditions, and approximations used to apply it. Classical mechanics, electromagnetism, thermodynamics, and quantum mechanics organize different aspects of these investigations. Their usefulness comes from turning measurements into predictions that can be checked and refined.

Consider a simple pendulum: how does its oscillation period change when its length changes? For small amplitude, the ideal model gives $T_0=2\pi \sqrt{\ell /g}$, where $g$ is gravitational acceleration. Taking $g=9.81m/{\mathrm{s}}^2$ as the model parameter, a one-metre pendulum has predicted period $2.0061\mathrm{s}$. Increasing its length to four metres gives $4.0121\mathrm{s}$: quadrupling the length doubles the period. The point mass cancels from the equation, so changing it alone does not change this model's answer.

The prediction identifies a mechanism and a way to challenge it. The small-angle approximation replaces $\sin \theta$ by $\theta$; the full ideal equation predicts a longer period at large amplitude. At $60^{\circ }$, its period is about $1.07318T_0$, already a $7.3\%$ change. Measurements at such amplitudes can reveal the limit of the approximation, while appreciable drag requires a damped pendulum. These distinctions are developed in simple pendulum and dimensional analysis. The small-amplitude model is also derived in Ling, Sanny and Moebs, University Physics, pendulums.

Several different experiments can become one calculation. Nondimensionalization rewrites the pendulum equation using $\tau =t\sqrt{g/\ell }$, leaving $d^2\theta /d{\tau }^2+\sin \theta =0$. All lengths share that dimensionless equation; the amplitude remains a parameter. Similarly, conservation laws and symmetry reveal features that many models share. This is one source of the subject's beauty: a compact relation can organize a large range of phenomena.

The connection to evidence remains essential when a model is elegant. In his introductory lectures, Richard Feynman emphasizes testing proposed relations by experiment and using idealizations whose accuracy depends on the conditions. ^[ref] A discrepancy can motivate a better measurement, a better approximation, or a different physical model; the mathematical calculation and the empirical assessment are both part of the explanation.

Positive feedback closes a causal loop whose net effect reinforces an initiating change. A loop with two inhibitory links can therefore have positive feedback. In signal transduction, a product that increases the activity of the enzyme making it illustrates reinforcement; the consequences depend on the full mathematical model, not just the loop sign. See signaling feedback.

A biological example is cell-cycle activation feedback. Cyclin B binds Cdc2, whose activity promotes activation of Cdc25; Cdc25 removes inhibitory phosphorylation from Cdc2, permitting further activation. Cdc2 also inhibits Wee1, an enzyme that inhibits Cdc2 through phosphorylation. That second return contains two inhibitions and therefore also has positive feedback. These are regulatory paths, not a claim that every connection is a single direct molecular encounter. ^[ref]

Reinforcement can preserve a decision after its initiating stimulus weakens: returning from the high state may require a lower input than entering it. The cell-cycle hysteresis experiment below tests this history dependence at matched cyclin B concentrations. A steep response alone is insufficient evidence of bistability, and finite observation times can make slow relaxation look like memory. A proposed switch must specify its input, readout and observation interval.

Relative centrifugal force is the dimensionless ratio of the local centrifugal acceleration magnitude $a$ to the conventional gravitational reference $g_0=9.80665\mathrm{m}{\mathrm{s}}^{-2}$. Despite its name, it is an acceleration ratio: the actual force on a specified mass $m$ is $ma$. With angular speed $\omega$ and rotor radius $r$, $a={\omega }^{2}r$. If $n$ is the numerical rotational speed in revolutions per minute and $r_{\mathrm{cm}}$ the numerical rotor radius in centimetres, unit conversion givesThus rpm alone does not specify the centrifugal acceleration.^[ref]

For a stipulated $3000\mathrm{rpm}$, rotor radii of $5$ and $10\mathrm{cm}$ give approximately $503$ and $1006$ times $g_0$. Doubling the rotor radius doubles relative centrifugal force; doubling rpm quadruples it. Even equal relative centrifugal forces do not guarantee equal separation: travel distance, elapsed time, viscosity, sample density and particle shape still matter. These are comparisons of a mathematical model, not instructions for running an instrument.

For an original idealized mathematical model, let $n_{{\mathrm{O}}_2}(t)$ be the amount of substance of molecular oxygen consumed since the initial reading and $n_{{\mathrm{CO}}_2}(t)$ the carbon dioxide produced. Suppose a fraction $f$ of that carbon dioxide is removed from the gas, all other gas inventories remain fixed, and temperature $T$ and gas pressure $P$ stay constant. The ideal gas law gives the positive gas contractionwhere $R$ is the molar gas constant and $V$ the gas volume. With complete absorption, $f=1$, the contraction reflects molecular oxygen uptake regardless of the amount of carbon dioxide made. For a uniform pipette bore of area $A$, an inward boundary displacement $x$ gives $D=Ax$. A graduated pipette can instead supply the displaced volume directly.

Let $D_s$ denote sample contraction and $D_b$ the matched blank contraction. Define $D_{\mathrm{corr}}=D_s-D_b$ with both taken relative to their own initial readings. Under the preceding assumptions,The slope of corrected volume against time estimates uptake over the chosen interval. The approximate sign acknowledges that a liquid boundary also responds to hydrostatic and capillary pressure, while the gas can exchange water vapour. A fixed-volume apparatus would instead show a pressure decrease; its ideal conversion is $n_{{\mathrm{O}}_2}\simeq V(P(0)-P(t))/(RT)$ after complete absorption and appropriate blank correction.

The ideal gas law also exposes drift without any biology: for fixed gas inventory, $V\propto T/P$. A one-percent rise in absolute temperature at fixed pressure produces a one-percent expansion. Matching the initial gas spaces makes such common changes easier to subtract. Subtracting unmatched blanks can leave a physical signal that looks biological.

The sedimentation coefficient $s$ is radial drift velocity divided by local centrifugal acceleration. It has units of time and combines particle properties with those of the surrounding liquid. In a dilute, deterministic mathematical model with negligible particle inertia, an effective displaced volume $V$, particle mass $m$, local liquid density ${\rho }_f$ and linear friction coefficient $f>0$ giveThe subtraction represents buoyancy. Equal mass does not imply equal $s$: changing shape changes drag force, and changing the medium changes both buoyancy and viscosity.^[ref]

For an isolated sphere of radius $b$ and uniform density ${\rho }_p$, Stokes's law supplies $f=6\pi \eta b$, where $\eta$ is dynamic viscosity, and $m=(4\pi /3)b^3{\rho }_p$. ThereforeThis original force-balance check predicts fourfold $s$ on doubling $b$ with everything else fixed, and zero drift when ${\rho }_p={\rho }_f$. It applies to dilute isolated spheres in Stokes flow, not arbitrary folded proteins or crowded suspensions. If $s$ and $\omega$ remain constant in a uniform medium, the ordinary differential equation $dr/dt=s{\omega }^{2}r$ has the exponential function solution $r(t)=r(0)e^{s{\omega }^{2}t}$. Arrival at an outer boundary therefore depends on starting position as well as $s$; a separation has no universal size cutoff.

Signaling adaptation is a decline of a response toward its earlier level while the initiating signal remains present. Specify which response is measured and its baseline: complete signaling adaptation means eventual return to that baseline; partial signaling adaptation leaves a lasting displacement. The continuing signal distinguishes signaling adaptation from simply switching off the stimulus. Signaling adaptation describes behavior, while signaling feedback describes a causal arrangement, so neither implies the other.

By reducing a persistent response to background ligand concentration, signaling adaptation can make a biological cell sensitive to changes rather than only to absolute levels. Resetting the response helps leave room for another change over a useful range of ligand concentrations. This is valuable in chemotaxis: swimming bacteria encounter changing chemical conditions, and their signal transduction influences whether a run continues or a tumble occurs. Signaling adaptation helps separate a change encountered during motion from a lasting background. This usefulness does not imply exact return or equal sensitivity at every possible concentration. ^{[ref] [ref]}

Here is a solvable incoherent feedforward loop toy that exhibits complete signaling adaptation without signaling feedback. The input $s(t)$ jumps from $0$ to $1$ at $t=0$ and remains $1$, a Heaviside step function. Let $x$ be a fast activating branch, $a$ a slower inhibitory branch, and $r=x-a$ a readout. The activities $s,x,a,r$ are dimensionless variables; physical time is in seconds. Choose time constants ${\tau }_x=1$ second and ${\tau }_a=5$ seconds, with initial conditions $x(0)=a(0)=0$:Both branches read $s$ directly. Neither branch depends on $r$, and $a$ does not change $x$: the subtraction only defines the readout. Thus the slow opposing influence is feedforward, with no closed feedback control loop.

For $t\ge 0$, solving the two ordinary differential equations givesWriting $t$ for the numerical time in seconds, these are $x=1-e^{-t}$, $a=1-e^{-t/5}$ and $r=e^{-t/5}-e^{-t}$. Initially $r=0$, then $x$ rises faster than $a$, producing a pulse. At long times $x$ and $a$ both approach $1$, so $r$ returns to $0$ despite sustained $s=1$. For this upward step $e^{-t/5}\ge e^{-t}$, hence $r(t)\ge 0$. The striking feature of this signaling adaptation toy is that both branches remain active while the readout becomes quiet: the continuing background and the response to its arrival are separable.

Find the pulse maximum by differentiating the exponential functions. With numerical time in seconds, the derivative isThe stationary condition is $e^{-4t/5}=1/5$, so the natural logarithm givesBefore this time the derivative is positive; afterwards it is negative. Together with $r(0)=0$ and ${\lim }_{t\to \infty }r(t)=0$, this proves that it is the unique pulse maximum. As another numerical check, at $t=10$ seconds $x\simeq 0.999955$, $a\simeq 0.864665$ and $r\simeq 0.135290$: the slow branch is still catching up.

The subtraction $r=x-a$ is a mathematical readout, not a claim that an actual chemical concentration equals this difference for arbitrary inputs. Nonnegativity was established only for the specified upward step; other input histories can give a negative signed readout. Complete signaling adaptation here also relies on equal eventual branch gains: changing the readout to $r=x-0.9a$ leaves $r_{\ast }=0.1$ for $s=1$. This toy illustrates cancellation and different time constants; it does not establish molecular implementation or robustness to arbitrary parameter changes. Ma and colleagues' 2009 study separately analyzed biochemical mathematical models with negative feedback and incoherent feedforward loops, identifying additional conditions for signaling adaptation in those models. ^[ref]

A historical connection to feedback control comes from Yi, Huang, Simon and Doyle's 2000 analysis of a bacterial chemotaxis model. They identified integral action in its regulation of receptor activity: accumulated deviations alter a slower adjustment, which in turn changes the measured activity. At a stationary state, an adjustment that integrates a nonzero error cannot remain constant, explaining how zero error can follow. Their conclusion depended on stated biochemical assumptions and stable behavior; it was not a claim that every biological cell uses that mechanism. A transient pulse alone, as the incoherent feedforward loop toy shows, cannot identify the underlying signaling feedback architecture. ^[ref]

Signaling feedback returns the effect of a downstream event to an earlier stage of signal transduction, closing a loop analogous to feedback control. Trace the causal arrows: if an earlier relay activates a later relay and the later relay changes the earlier relay's activity, the return path closes the loop. Negative feedback opposes the initiating change; positive feedback reinforces it. Signaling feedback is a possible property of signal transduction, not a requirement of every pathway.

For a schematic example, let relay $A$ activate relay $B$. If $B$ inhibits $A$, increased $A$ eventually produces an opposing influence: negative feedback. If $B$ activates $A$, increased $A$ produces a reinforcing influence: positive feedback. These signs describe effects on activity, rather than the direction in which molecules travel. A closed loop containing two inhibitions can have positive feedback, because inhibiting an inhibitor reinforces the original change. The sign belongs to the whole loop.

Negative feedback can restrain excessive activation and help preserve a usable range of responses. Positive feedback can support a decisive transition: Alberts and colleagues describe an enzyme whose product increases its own activity. Such reinforcement can contribute to switchlike behavior, but a positive feedback arrow alone does not prove bistability. A different structure, an incoherent feedforward loop, sends opposing influences forward from the same input; inhibition there is not a downstream return. ^{[ref] [ref]}

A worked mathematical model shows why negative feedback does not automatically produce complete signaling adaptation. Let $s$ be a fixed normalized input, $r$ a normalized response, $g\ge 0$ a dimensionless feedback gain, and $\tau >0$ a time constant. Suppose response removal contributes $-r$ and a return proportional to the response contributes $-gr$. The ordinary differential equation isSetting the derivative to zero gives the equilibrium point $r_{\ast }=s/(1+g)$. A displacement $\delta r=r-r_{\ast }$ obeys $\tau d(\delta r)/dt=-(1+g)\delta r$, so its decaying exponential function establishes asymptotic stability. For a unit input switched on at $t=0$, initial condition $r(0)=0$, $g=2$ and $\tau =1$ second,The negative feedback reduces the eventual response from $1$ to $1/3$, but this response rises monotonically and never returns to its zero baseline. Attenuation and complete signaling adaptation are different properties. This deliberately simple mathematical model specifies no molecular pathway; integral action provides a contrasting way to remove a lasting offset in a stable feedback control system. ^[ref]

The SIR model is a mathematical model of an epidemic that follows three groups: a susceptible compartment whose members can acquire infection, an infectious compartment whose members can pass it on, and a removed compartment whose members neither acquire nor pass on infection. The letters stand for susceptible, infectious and recovered or removed. Here we study the simplest closed, constant-parameter version, with homogeneous mixing and no births, deaths, migration or waning immunity. A person who leaves the infectious compartment never returns to the susceptible compartment in this version.^[ref]

The appeal of the SIR model is that one can explain a rise and fall in the infectious compartment without changing anyone's behavior: fewer opportunities for new infection eventually reverse the growth. The SIR first integral even determines the SIR infection peak without computing its time. These are useful deductions from explicit assumptions. The stipulated numbers below illustrate the mathematical model; they are neither estimates for an actual disease nor a forecast or medical advice.

Kermack and McKendrick's 1927 paper, "A contribution to the mathematical theory of epidemics", in Proceedings of the Royal Society A, is an early foundation for the SIR model. Its DOI is 10.1098/rspa.1927.0118, also recorded in the Royal Society's archive of contemporary referee reports.^[ref][ref]

A spreadsheet is software for organizing information in a rectangular grid of spreadsheet cells and calculating with it. Rows are numbered and columns are lettered: B3 is the spreadsheet cell at column B, row 3. A spreadsheet cell can hold a label, a numerical input, or a spreadsheet formula whose value supplies an output. This brings a mathematical model close to its data: quantities, assumptions and results can be read side by side. The address convention is described in LibreOffice's official help. ^[ref]

The useful idea is to write a relationship once and reuse it across similar rows. Relative spreadsheet references follow each row's data; absolute spreadsheet references retain a shared assumption; mixed spreadsheet references let one coordinate vary while the other stays fixed. Spreadsheet dependency evaluation then computes the resulting spreadsheet formulas. A rectangular layout organizes the model for its reader, while the spreadsheet dependency graph organizes its calculation.

The susceptible compartment consists of people who can acquire infection. In the closed SIR model, $S(t)$ is their count and $s(t)=S(t)/N$ their fraction of the fixed total $N$. Membership expresses the assumptions of the mathematical model, rather than measuring the strength of an individual's immunity.