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A histone H1 kinase assay measures protein kinase action on supplied histone H1, a protein substrate. When the transferred phosphate carries a radioactive isotope label, label incorporated into histone H1 reports catalytic action over the assay interval. It does not count CDK1 molecules, and a crude preparation can contain other protein kinases acting on histone H1. Attribution therefore needs controls.

A 2010 study prepared human HeLa S3 extracts after a double thymidine block, using swelling, a Dounce homogenizer, centrifugation, dialysis and concentration. Added recombinant cyclin B1 incubated for $20$ minutes.^[ref]

Aliquots received histone H1 and terminal-${}^{32}\mathrm{P}$-labeled ATP for a $15$-minute room-temperature reaction, stopped with Laemmli buffer. Dried SDS-PAGE gels exposed a phosphorimager screen; a BioRad Molecular Imager FX quantified labeling. No-added-cyclin B activity set the normalization baseline; RO-3306 reduced H1 activity approximately $85\%$. Companion Western blots monitored Tyr15 phosphorylation, with actin Western blot loading control.^[ref]

Mixed extracts lose compartment boundaries; inhibitor sensitivity supports a major CDK1 contribution, with residual signal.^[ref] Substrate choice and inhibitor specificity limit attribution. The cell-cycle hysteresis experiment supplies a separate history comparison with this kind of histone H1 readout; its dose thresholds should not be transferred to human biological cells.

Homeostasis is the maintenance of usable internal conditions despite disturbances. It concerns selected regulated physiological variables, rather than keeping every part of an organism unchanged: core body temperature may remain near its usual range while skin temperature, sweating and blood pressure change. Regulation makes a relatively dependable environment for biological cells out of a changing external world. ^[ref]

A useful causal description separates a physiological sensor, a physiological integrator and a physiological effector. The return from the physiological effector's action to the sensed regulated physiological variable closes a feedback control loop. In negative feedback, that return opposes the disturbance. The insulin and glucagon articles already trace the blood glucose example; synaptic integration explains how several inputs can jointly influence a neuron. These are implementations and ingredients, not a requirement that every regulatory loop use one anatomical control center. ^[ref]

Homeostasis is useful precisely because internal activity can change: a nearly steady temperature can conceal a large increase in heat transfer. Conversely, observing a flat trace alone does not identify a regulatory mechanism. The environment might simply be constant, or the measuring instrument too insensitive; a perturbation and an independent response readout provide stronger evidence. The skin-temperature perturbation experiment below makes that distinction tangible.

The historical motivation was the puzzle of stability in changing living material. In his 1932 The Wisdom of the Body, Walter Bradford Cannon emphasized that an organism exchanges material with its surroundings and continually replaces its own structure, yet maintains relatively steady internal conditions. This authored excerpt presents homeostasis as active compensation for external and internal disturbances, rather than a static body. ^[ref]

Here is an explicitly synthetic mathematical model of a warming disturbance and its compensatory cooling. It is a teaching analogy for thermoregulation, not a measured human heat budget. Write $x=(T-T_r)/\Delta T$, where $T$ is a lumped internal temperature, $T_r$ is a fixed reference and $\Delta T>0$ is a chosen temperature scale. Let $d$ be an additional heating rate and $u$ an additional cooling rate, both scaled to the same reference rate. All three are dimensionless variables, while $\tau >0$ is a time constant in the chosen time units. The ordinary differential equation isThe term $-x$ idealizes increased passive heat transfer with increased internal temperature, around the baseline. The term $-u$ opposes warming through proportional control with feedback gain $k$. This local signed model does not treat a negative $u$ as negative sweating: it would represent decreased cooling relative to baseline. Changes of the environment and multiple body compartments are omitted.

For a constant disturbance $d=D$ and initial condition $x(0)=0$, the exponential function solution isSubstitution checks the ordinary differential equation directly. Increasing $k$ reduces the lasting offset and, in this instantaneous model, shortens the response time constant to $\tau /(1+k)$. It does not make $x_{\ast }$ zero for a nonzero disturbance and finite $k$: a persistent cooling command requires a persistent displacement under proportional control. At the equilibrium point, the extra heating is balanced by passive loss $x_{\ast }$ plus active cooling $u_{\ast }$. Zero change of temperature means balanced rates, rather than zero rates, consistent with conservation of energy.

The first diagram uses $D=1$, $\tau =4$, and a pulse from $t=2$ to $t=10$, comparing $k=0$ with $k=3$. The latter approaches $x_{\ast }=1/4$ during the pulse, compared with $1$ without active feedback control. After the pulse ends, each displacement decays as $x(10)e^{-(1+k)(t-10)/\tau }$. These are computed curves, not physiological measurements. The $k=0$ curve still has passive loss; it is not a claim that an unregulated body warms without limit.

Real physiological effectors take time to respond. To isolate that issue, set $\tau =1$ and replace the instantaneous command by another ordinary differential equation:Here the disturbance remains $1$ from $t=0$ onward and the effector time constant is also $1$. The equilibrium point is still $(x_{\ast },u_{\ast })=(1/5,4/5)$, but the characteristic polynomial is $(\lambda +1{)}^2+4$. Its eigenvalues $-1\pm 2i$ give decaying oscillations and asymptotic stability. The exact response isIn particular, $\dot{x}=e^{-t}\cos (2t)$, so the first maximum occurs at $t=\pi /4$, with $x=1/5+(2/5)e^{-\pi /4}\simeq 0.3824$. The response exceeds its eventual displacement while remaining above its baseline. The return has the same negative sign as before; the extra state changes the timing. A finite effector time constant is distinct from the fixed time lag in the delayed negative feedback model.

Maintenance of a relatively constant body temperature. It differs from the metabolic source of heat described by endothermy.

homeoviscous adaptation changes membrane composition in response to conditions such as growth temperature, helping preserve useful membrane dynamics. The original Escherichia coli study measured temperature-dependent fatty-acid changes and membrane-lipid viscosity by spin labeling.^[ref]

Use the isothermal Lane-Emden equation variables $r=a\xi$, $a^2=c_T^2/(4\pi G{\rho }_c)$ and $\rho ={\rho }_c{e}^{-\Psi }$. Truncation at fixed dimensionless radius ${\xi }_b$ gives $R=a{\xi }_b$ and $M=4\pi {\rho }_c{a}^3{\xi }_b^2{\Psi }^{\prime }({\xi }_b)$. Division proves the displayed relation. A common temperature, composition and dimensionless truncation therefore imply $M\propto R$, while the external pressure generally varies along that homologous sequence. Arbitrary changes of truncation at fixed temperature do not retain a common coefficient. A finite regular cloud needs an outer boundary such as that of a pressure-confined isothermal sphere.

For finite-radius regular polytropes of one fixed index $n$ and composition, let ${\xi }_1$ be the first zero of the Lane-Emden function and $Y_n=-{\xi }_1^2{\theta }^{\prime }({\xi }_1)$. Hydrostatic scaling gives $GM/R=(n+1)(Y_n/{\xi }_1)\mathcal{R}{T}_c$. Fixed central temperature therefore implies $M\propto R$. This is not the fixed-entropy-constant sequence; if the index or composition changes, their prefactors cannot be discarded.

The absolute temperature divided by an appropriate melting temperature, $T/T_m$. It is a useful scale for comparing thermally activated deformation; Celsius temperatures cannot be used directly in the ratio.

Horizontal convection is buoyancy-driven circulation forced by lateral variations of temperature along one horizontal boundary. Hot and cold surface regions drive a circulation even though all thermal forcing is applied at the same height. An insulated bottom and sides require zero net heat exchange through the remaining surface in steady state, but permit heat to enter its warmer region and leave its colder region.

For a unit-depth Boussinesq layer with stress-free boundary conditions, fixed boundary temperatures, imposed field along $y$, and horizontal magnetic perturbations at the boundaries, put $a_h^2=k^2+m^2$ and $\Lambda =a_h^2+{\pi }^2$. Poloidal vertical velocity, temperature and vertical field proportional to $e^{st+ikx+imy}\sin \pi z$ have amplitudes satisfyingThe determinant gives the stated dispersion relation. Here $\sigma$ is the Prandtl number, $\zeta$ the magnetic-to-thermal diffusivity ratio, $Q$ the Chandrasekhar number and $R$ the Rayleigh number. The steady neutral curve is $R={\Lambda }^3/a_h^2+Q{m}^2\Lambda /a_h^2$.

// Target: fluid-mechanics.bigb

Dilute shock-heated interstellar plasma with characteristic temperatures around $10^6\mathrm{K}$. supernova energy injection and subsequent cooling are important to its evolution; temperature alone does not determine its volume filling factor.

Consolidation under approximately equal applied pressure in all directions at elevated temperature. In waste treatment it can produce a dense ceramic inside a containment can.

A porous sheet of thickness $d$ and permeability of a porous medium $\Pi$ supplies a thin premelting film. The Darcy flow law equates water supply to ice growth for equal densities. With equal film and ice conductivities, negligible latent heat relative to the through-going conductive flux, and $a\ll h$, phase equilibrium gives $p_T=(\rho L\Delta T/T_m{)}^{3/4}(A/(6\pi h^3){)}^{1/4}$. The increasing gravitational load limits growth. A quasi-steady temperature field alone does not imply the negligible-latent-heat approximation.

Hydrodynamic atmospheric escape is a collisional transonic outflow in which bulk gas carries multiple species away. It replaces the hydrostatic Jeans-tail description when gravity is weak compared with thermal energy.

The hydrodynamic radius is the radius of a hypothetical no-slip sphere with the same translational diffusion coefficient as the dissolved species. In the dilute continuum regime the Stokes–Einstein relation giveswhere $k_B$ is the Boltzmann constant, $T$ the absolute temperature, $\eta$ the dynamic viscosity of the solvent, and $D$ the translational diffusion coefficient. This effective radius need not equal the radius of gyration or a geometric molecular radius. It is a useful descriptor for SEC molar mass calibration, rather than a complete specification of how every shape enters every pore.

Partial hydrogen ionization changes disk cooling and permits cold and hot stable thermal branches separated by unstable equilibria. A dwarf nova can cycle between a cold matter-storing disk and a hot, rapidly accreting disk. The cycle is described by the evolution of local surface density and temperature, coupled to viscous evolution of an accretion disk.

Let $u_s=⟨ϵ{⟩}_s$ and $v_s={\mathrm{Var}}_s(ϵ)$ for the conditional distributions of parahydrogen and orthohydrogen, with $s=\mathrm{p},\mathrm{o}$. Differentiating the normalized Boltzmann factors gives $d{u}_s/dt=v_s/t^2$. Thus the rotational heat capacity at constant volume per molecule of a composition frozen at $x$ isAt conversion equilibrium, the law of total variance adds a nonnegative contribution from the changing species populations:The last term is the contribution from conversion. Compare the first two terms with a frozen sample at the same instantaneous composition; a frozen $3:1$ sample generally has different weights too. This formula follows directly from the variance of a mixture: subtract the overall mean inside each species, expand the square, and the cross terms vanish because each conditional deviation has zero mean. It also proves nonnegative rotational heat capacity in either model.

For an original worked example at $t=1/2$, summing the allowed ladders gives $q_{\mathrm{p}}\approx 1.00003072$, $q_{\mathrm{o}}\approx 0.05494692$, and $x_{\mathrm{o}}^{\mathrm{eq}}\approx 0.14150982$. The calculated rotational heat capacities are $c_{\mathrm{eq}}/k_{\mathrm{B}}\approx 1.94719633$ and $c_{\mathrm{fr}}/k_{\mathrm{B}}\approx 0.00110733$ for frozen normal hydrogen. The contrast is large because changing species accesses the $J=0$ to $J=1$ separation, whereas fixed species must use the larger $J=0$ to $2$ or $J=1$ to $3$ separations. These are model numbers at a chosen dimensionless temperature, not measured hydrogen data.

Both rotational heat capacities approach zero as $t\to 0$, although frozen normal hydrogen retains nonzero rotational internal energy. Both approach $k_{\mathrm{B}}$ as $t\to \infty$ within the rotor model. The high-temperature limit is a formal limit with all other excitations suppressed; actual molecular hydrogen eventually needs molecular vibration, nonrigidity, and dissociation. Multiplying the per-molecule result by the Avogadro constant gives rotational molar heat capacity in units of the molar gas constant $R$. A dilute classical ideal gas additionally contributes $3R/2$ from translation to its heat capacity at constant volume.

A floating density instrument. At equilibrium, Archimedes' principle gives $\rho V_{\mathrm{displaced}}=m$ for fixed instrument mass $m$, so it displaces more volume in a less dense liquid. A graduated narrow stem converts this volume change into a visible immersion reading. Calibration specifies a temperature and a reading convention for the meniscus; a sample's other dissolved substances can affect a sugar-calibrated scale.

The hydrophobic effect is the solvent-mediated tendency of nonpolar regions to associate in water. Exposing a hydrocarbon region changes how surrounding water molecules can arrange and interact; reducing that exposure can lower the total free energy. Both water and lipids enter the accounting. The effect is not an additional covalent bond between chains. See IUPAC Gold Book: hydrophobic interaction.

At fixed temperature and pressure, compare two candidate organizations by the change in Gibbs free energy, $\Delta G=\Delta H-T\Delta S$, where $H$ is enthalpy, $T$ is absolute temperature, and $S$ is entropy. An organization with lower $G$ is favored in this comparison. Reducing exposed hydrocarbon area is one contribution; changes in chain packing and the entropy of the lipids also matter. This explains why hydrophobicity alone cannot predict every aggregate shape. See 16.4 Free Energy.