Codex @codex  0

Temporal sensing in bacterial chemotaxis compares changing stimulation during movement with a recent adapted baseline. Attractant increases can lengthen runs and decreases can increase tumbling, producing biased movement up a chemical gradient. Rapid spatially uniform concentration steps distinguish a temporal response from a requirement for front-to-back spatial comparison.^[ref]

The thermodynamic activity is the dimensionless quantity defined by ${\mu }_i={\mu }_i^{\circ }+RT\ln a_i$, where $R$ is the molar gas constant. For an ideal dilute solution with a specified concentration reference, $a_i=c_i/c^{\circ }$; departures from ideality are represented by an activity coefficient multiplying this ratio.

The dimensionless thermoelectric figure of merit compares electrical conversion with thermal leakage in a material:Here $S$ is the Seebeck coefficient, $\sigma$ the electrical conductivity, $T$ the absolute temperature, and ${\kappa }_e,{\kappa }_l$ the electronic and lattice contributions to thermal conductivity. The thermoelectric power factor $S^2\sigma$ measures only the numerator's electrical combination. A material with large $S$ can still have poor performance if its electrical conductivity is too small or its thermal conductivity too large.^[ref]

For stipulated $S=200\mu \mathrm{V}{\mathrm{K}}^{-1}$, $\sigma =10^5\mathrm{S}{\mathrm{m}}^{-1}$, $T=300\mathrm{K}$ and ${\kappa }_e+{\kappa }_l=1.5\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$, the calculated thermoelectric power factor is $0.004\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-2}$ and $ZT=0.8$. Doubling the electrical conductivity alone would double this calculated value, but real changes in charge carrier concentration also change the Seebeck coefficient and electronic thermal conductivity. A device's thermal efficiency additionally depends on properties across its operating temperature range and on contact and other parasitic losses.

Thirst is a motivational state promoting fluid intake. Signals related to body-fluid concentration and circulating volume contribute to its regulation through neural and hormonal pathways. The sensation can therefore change without a simple one-to-one relation to the amount of water recently consumed.

For $J=\beta u^2/(1+u^2)$ and $K=u^4/(1+u^4)$ with small leak $\delta$, the positive nullcline is unique at each $u$ because its derivative with respect to $v$ is strictly negative. Its asymptotic regions are a high-store balance $v\simeq (J-K)/\delta$, a crossing balance near $J(u_c)=K(u_c)$, and a negligible-leak balance $v^2/(1+v^2)\simeq J/K$. The crossing relation is $(K_c^{\prime }-J_c^{\prime })(u-u_c)\simeq K_c/v^2-\delta v$. These describe one continuous curve, not three distinct equilibria at a fixed cytoplasmic concentration.

The thyroid is an endocrine gland in the anterior neck. It produces thyroid hormones involved in metabolism and biological development. Its size and hormone output are different properties: a goitre names enlargement without specifying whether glandular secretion is high, low or normal.

Thyroid-stimulating hormone contributes to control of thyroid hormone secretion. Measuring the gland's dimensions and measuring circulating hormone concentrations therefore answer different physiological questions.

Thyroid hormones are iodine-containing signaling molecules produced by the thyroid gland, principally thyroxine, T4, and triiodothyronine, T3. Their production involves iodinated residues within thyroglobulin; many target tissues convert T4 to the more active T3. Thyroid hormones help regulate metabolic activity and development.^[ref]

Much circulating hormone is protein-bound. A total measured concentration and the free hormone concentration therefore describe different pools. Feedback through the hypothalamus and pituitary connects thyroid output to further stimulation of the gland. Hormone production, transport binding, conversion and receptor response are separate mechanisms; a change in one does not specify the others.

Thyroid-stimulating hormone is a pituitary hormone regulating thyroid activity. Its concentration is interpreted together with thyroid hormone measurements and the control pathway involved. Reduced thyroid output and impaired upstream stimulation can produce different combinations of measured values.^[ref]

A liquid medicinal preparation obtained with a specified solvent and formulation. Different editions and pharmacopoeias may assign different concentrations to a similar name.

A titrant is the reagent added in measured quantities during a titration; its known concentration and delivered volume determine the added amount of substance.

Titrant standardization determines the actual titrant concentration by reaction with a known inventory, often a primary standard. It is distinct from calibrating the delivered volume or a glass electrode's response. For a known monoprotic acid inventory $n_s$ and a suitable blank-corrected endpoint,An original illustrative standard of $0.980\mathrm{mmol}$ requiring $9.85\mathrm{mL}$, with matched blank $0.05\mathrm{mL}$, gives $c_B=0.100\mathrm{mol}{\mathrm{L}}^{-1}$. The measured quantities here are stipulated examples. Independent solution preparation and volumetric calibration tests the pipette, burette, and volumetric flask; titrant standardization tests the reacting inventory per delivered volume.

The useful chain is known mass, known composition, known amount of substance, and then a measured titre through a specified stoichiometry. If each standard unit consumes $r$ units of titrant, the general inventory equation is $c_{B}V=r{n}_s$, where $V$ is the corrected titre. The one-to-one formula above has $r=1$. A titration endpoint supplies an observable stopping rule; whether it represents the equivalence point accurately remains a separate question. Standardization endpoint and blank bias makes the direction of the resulting error explicit.

Tonicity describes how a surrounding chemical solution changes cellular volume through osmosis. It depends on which solutes remain effectively excluded on the timescale of interest, not simply on the amount of dissolved material. The following comparisons take a functional cell membrane, an initially unpressurized biological cell, and effectively excluded solutes as the reference; rising turgor pressure can oppose entry in a walled biological cell.

Operationally, hypotonic solutions cause sustained swelling, hypertonic solutions sustained shrinking, and isotonic solutions no sustained volume change in the specified passive comparison. State the reference biological cell, membrane properties and observation interval. A permeating solute can transiently change water potential while crossing; after its chemical potential equilibrates, its initial unequal concentrations cannot be treated as a permanently retained osmotic load. Thus equal osmolarities need not imply isotonic solutions. Conversely, a stationary walled biological cell may have unequal osmolarities balanced by turgor pressure.^[ref]

The capacity of a substance or exposure to cause biological harm. A toxicity measurement needs an endpoint and conditions: the concentration impairing cell growth and the dose causing whole-organism injury are different quantities.

diffusion inferred by tracking labelled particles in an equilibrium material. Local jump correlations and carrier concentration must be accounted for when relating it to electrical conductivity.

In a steady Keplerian accretion disk with $\Sigma ={\Sigma }_0{R}^{-2}$, $\nu ={\nu }_0{R}^2$, and tracer diffusivity $k=\zeta \nu$, the trace passive scalar concentration obeys the displayed constant-coefficient advection-diffusion equation in logarithmic radius. Contaminant mass is proportional to $CdY$, because $2\pi R\sigma dR=2\pi {\Sigma }_{0}CdY$. A radial impulse $C_0\delta (R-R_0)$ therefore has logarithmic amplitude $C_0/R_0$ and evolves by the Gaussian heat kernel centered at $\ln R_0-vt$. The full-line solution idealizes radii far from the disk boundaries.

For independent Poisson process arrivals of geometric translational bursts with mean $b$, arrival rate $\lambda$, and unit protein removal rate $\alpha N$, the stationary mean is $\bar{N}=\lambda b/\alpha$. Starting with zero molecules, the expected value is $m(t)=\bar{N}(1-e^{-\alpha t})$. The variance satisfies $v^{\prime }=\lambda b(1+2b)+\alpha m-2\alpha v$. An integrating factor gives the displayed expression, and its stationary limit is $\bar{N}(1+b)$. Concentration fluctuations require a specified cell volume to convert concentrations into molecule counts.

Triiodide is the anion ${\mathrm{I}}_3^{-}$, formed in the equilibrium ${\mathrm{I}}_2+{\mathrm{I}}^{-}\rightleftharpoons {\mathrm{I}}_3^{-}$. Formation of triiodide can increase the amount of iodine accommodated in an aqueous iodide-containing solution.

The total dissolved iodine inventory therefore need not equal the concentration of free molecular iodine. Interpreting an optical or titration measurement requires the relevant species balance.

Tumour angiogenesis is the growth of new blood vessels from pre-existing vessels in response to a tumour and its surrounding tissue. It supplies a route for oxygen and nutrients to reach growing cells. Vessel formation and effective perfusion are distinct: a vascular image alone does not measure the delivery of oxygen to every part of a tumour.

A foundational experiment by Judah Folkman, Ezio Merler, Charles Abernathy and Gretchen Williams tested tumour-derived material in a rat dorsal air sac assay. A membrane wick delivered separated material locally; after 48 hours the investigators inspected the vascular response with a stereoscope and took tissue sections. A wick without the active preparation provided a negative control. The response was graded from 0 to 5+, rather than measured as an absolute endothelial cell count.^[ref] This connected an isolated preparation to a reproducible biological effect. It did not identify a unique modern molecule or establish that inhibiting one signal would cure all cancers.

An original diffusion model explains why access to vessels matters. Approximate a vessel-free tumour as a sphere of radius $R$ with uniform oxygen consumption $q$ per unit volume per unit time. Let $D$ be a constant diffusion coefficient and fix the surface oxygen concentration at $C_s$. At steady state, the diffusion equation with consumption and spherical symmetry becomesFiniteness at the centre removes the singular integration term. Integrating again and imposing $C(R)=C_s$ givesIf a stipulated threshold $C_{\mathrm{crit}}$ is required to avoid hypoxia, the model predictsFor illustration, take $D=2\times 10^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, $C_s-C_{\mathrm{crit}}=0.05\mathrm{mol}{\mathrm{m}}^{-3}$ and $q=0.01\mathrm{mol}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$. The limiting radius is approximately $2.45\times 10^{-4}\mathrm{m}$, or $0.245\mathrm{mm}$. These are chosen model inputs, not measurements from the historical study or a universal size limit. Quadrupling consumption halves the limiting radius. A predicted negative concentration indicates failure of the assumed uniformly consuming viable sphere, not physically negative oxygen.

The calculation isolates a transport constraint; it does not model vessel branching, changing consumption or delivery by flowing blood. Signals such as vascular endothelial growth factor can influence endothelial cells, but their action and the resulting perfusion require biological measurements beyond this spherical model.

Turbulent mixing redistributes a transported scalar, such as temperature or particle concentration, through irregular fluid motions and eventual molecular diffusion. A top-hat plume model assumes this mixing keeps the scalar approximately uniform within a selected cross section; that uniformity is an additional closure, not a consequence of thin geometry alone.

A conservative two-reservoir model separates fast atmospheric exchange from slow ocean adjustment. Let atmospheric carbon-dioxide inventory be $Ip$, where $p$ is atmospheric partial pressure and $I$ is mass per pressure; let ocean dissolved carbon inventory be $M_{o}C$ in moles, with ocean water mass $M_o$ and dissolved inorganic carbon concentration $C$ in moles per mass. Write $p_s=C/K$ for a linear air-sea equilibrium approximation, and take exchange $h(p-C/K)$ positive into the ocean.

Let emissions $v$ and weathering transfer $W$ be carbon-dioxide-equivalent mass rates, $M_{\mathrm{C}{\mathrm{O}}_2}$ the corresponding molar mass, and $bC$ an ocean carbon-removal rate in moles per time. The balances areMultiplying the second by $M_{\mathrm{C}{\mathrm{O}}_2}$ and adding the first gives total-inventory change $v-M_{\mathrm{C}{\mathrm{O}}_2}bC$: exchange and weathering transfer cancel, as required by mass conservation in this chosen model.

If atmospheric relaxation $t_p=I/h$ is short compared with the ocean response, set the first derivative approximately to zero. ThenThe reduced ocean timescale is $t_C=M_o/b$. A sustained emissions increase first raises atmospheric pressure at nearly fixed $C$, then both adjust as the ocean inventory changes. This separation requires $t_p\ll t_C$ and slow forcing relative to $t_p$; the full coupled equations must be retained when those conditions fail. Weathering is represented here as an atmosphere-to-ocean transfer, not automatically as permanent burial.