A solvated electron in water, conventionally $e_{\mathrm{aq}}^{-}$. Its lifetime and reactions depend on scavengers, acidity and dissolved gases.

The Earth's outer core is liquid. P waves slow strongly across the mantle-to-outer-core transition, while ordinary shear waves do not propagate through the liquid. This produces major changes in seismic ray paths and shadow ranges.

The Arabidopsis kinase also called SnRK2.6, important for ABA-dependent control of guard cell transport. Its activity is restrained by selected protein phosphatase 2C enzymes.

Størmer's theorem states that for any fixed finite set $S$ of primes, there are only finitely many pairs of consecutive positive integers whose prime factors all belong to $S$. Equivalently, the equation $y-x=1$ has only finitely many solutions in positive $S$-smooth integers.

For $S=\{2,3\}$, the pairs include $(1,2),(2,3),(3,4)$ and $(8,9)$. The underlying method reduces the search to finitely many Pell equations by separating square factors from the products of allowed primes. The allowed prime set must stay fixed; allowing it to grow with the candidate integers would remove the finiteness assertion.

A 5-smooth number, also called a Hamming number, is a positive integer of the form $2^a{3}^b{5}^c$ with $a,b,c\ge 0$. The sequence begins $1,2,3,4,5,6,8,9,10,12,15,16,18,20,\dots$.

The number seven is absent, while arbitrarily large powers of two are present. To generate the sequence in increasing order, merge the three sorted sequences formed by multiplying earlier values by $2,3,5$, removing duplicates. For a finite bound $B$, an equivalent search enumerates exponent triples satisfying $a\log 2+b\log 3+c\log 5\le \log B$.

A 47-smooth number is a positive integer whose prime factors are all at most $47$. Equivalently, it is a product of nonnegative powers of $2,3,5,7,11,13,17,19,23,29,31,37,41,43,47$. The number one is included as the empty product.

For example, $47\cdot 2^{10}$ is 47-smooth, while $53$ is not. The bound constrains the prime factors rather than the size of the number. Such classes make multiplicative searches manageable by replacing an arbitrary factorization with an exponent vector over a fixed finite prime set.

Sar1 GTPase is a small GTP-binding protein that initiates COPII coat assembly at the endoplasmic reticulum. Its GTP-bound state associates with the membrane and recruits the Sec23-Sec24 complex, followed by the Sec13-Sec31 complex. Nucleotide cycling helps coordinate coat assembly and turnover. Reconstitution with defined components demonstrates a direct role in carrier formation rather than only an indirect consequence of cellular signalling. ^[ref]

An ARF GTPase is a small GTP-binding protein involved in membrane organization and coat recruitment. Nucleotide exchange promotes a membrane-associated active state that recruits selected effectors, while hydrolysis contributes to turnover of the assembly. ARF1 is a prominent regulator of coatomer recruitment at Golgi-associated membranes. Different family members and effectors act in different locations, so the family name does not specify one universal cargo route. ^[ref]

A Rab GTPase is a small GTP-binding protein that regulates membrane traffic. GDP- and GTP-bound states differ in their interactions with effectors, while exchange factors and GTPase-activating proteins regulate the cycle. Different Rabs act at different trafficking stages, helping organize compartment identity, motor recruitment and carrier targeting. Rab8a and Rab11a illustrate related switches with different preferred routes and partners. ^[ref]

A small GTPase participating in protrusion and polarity in migrating cells.

Nucleotide exchange changes recruitment of effectors that regulate the actin cytoskeleton. Spatially restricted activity can help select a protrusive region, connecting a biochemical switch to directional cell movement. ^[ref]

Size-exclusion chromatography with multi-angle light scattering combines separation by hydrodynamic behaviour with light-scattering and concentration measurements. With appropriate optical parameters it estimates molar mass across a peak, helping distinguish assembly from shape-dependent elution.

The unfolded site frequency spectrum counts segregating sites according to the number of sampled copies carrying the derived allele. In a sample of five haploid sequences, a mutation carried by one sequence contributes to the singleton class, while one carried by four contributes to class four. Assigning ancestral and derived states is essential: reversing them exchanges classes $i$ and $n-i$. A folded site frequency spectrum removes this directional distinction. ^[ref]

For a simple single-tether geometry, imposed displacement $Z$ divides into cantilever deflection $d$ and molecular extension $x$. Applying the displayed correction prevents attributing the instrument's stretch to the molecule. Geometric offsets, handles and substrate compliance require additional calibrated terms.

A measurement protocol that holds applied force approximately constant while allowing displacement to vary. Feedback can adjust an actuator in response to a calibrated load signal; alternatively, a passive field gradient can be sufficiently uniform over the displacement range. Finite feedback bandwidth, drift and spatial force variation limit the approximation. This differs from fixing actuator position or prescribing a force ramp.

An inferred ordering of biological cells by a model of state progression. It is not by itself a measured time, a lineage trace or proof of causal developmental transitions.

An inferred ordering of molecular states along a developmental trajectory. It does not itself establish which cell divided to produce another cell.

For support displacement $\mathbf{a}(t)$ and angle measured from the downward vertical, the mass position is $(a_x+l\sin \theta ,a_y-l\cos \theta )$. Its Euler-Lagrange equation is $l\ddot{\theta }+{\ddot{a}}_x\cos \theta +(g+{\ddot{a}}_y)\sin \theta =0$. In the support frame the effective acceleration is $\mathbf{g}-\ddot{\mathbf{a}}$; taking its torque about the hinge gives $l^2\ddot{\theta }\hat{\mathbf{z}}=\mathbf{l}\times (\mathbf{g}-\ddot{\mathbf{a}})$. The prescribed support can do work, so pendulum energy need not be conserved.