RNA melting temperature ($T_m$)

An RNA melting temperature, $T_m$, characterizes the midpoint of a specified RNA conformational transition under stated conditions. In a reversible, unimolecular two-state RNA melting model, it is the temperature at which folded and unfolded populations are equal. A midpoint assigned to an optical trace requires a model connecting absorbance to those populations.

Let unfolding free energy be $\Delta G_u=G_U-G_F$. If unfolding enthalpy and entropy are treated as temperature independent,The formula assumes a finite relevant transition and uses absolute temperature. With original stipulated values $\Delta H_u=100\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ and $\Delta S_u=300\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$, $T_m\approx 333.3\mathrm{K}$. Holding enthalpy fixed but changing entropy to $310\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$ gives approximately $322.6\mathrm{K}$. A melting temperature reflects both energetic and entropic contributions.

Molecularity matters. An ideal intramolecular hairpin transition need not change its midpoint with total strand concentration. A transition involving association of separate strands generally does: the equilibrium relation includes their concentrations. The concentration dependence of RNA melting can therefore help test which structural model is appropriate. In the 2010 tetraloop measurements, concentration-independent curves across a broad concentration range supported stem–loop formation.^[ref]

A raw halfway absorbance is not automatically this thermodynamic midpoint. If the folded and unfolded RNA melting baselines at one temperature are $0.55$ and $0.75$, an absorbance of $0.60$ implies unfolded fraction $0.25$, not $0.50$. Intermediates, temperature-dependent parameters or RNA melting hysteresis can further complicate midpoint assignment. Report the solution conditions and definition of the measured transition rather than treating $T_m$ as a context-free constant of an RNA sequence.

The concentration dependence of RNA melting is the change of an RNA melting temperature or population curve with total strand molar concentration, while other solution conditions are held fixed. It helps distinguish intramolecular RNA hairpin melting from association of separate strands, provided the system reaches chemical equilibrium.

For an ideal unimolecular folded-to-unfolded transition, $F\rightleftharpoons U$, the equilibrium constant is $K_u=[U]/[F]$. Multiplying both populations by the same amount leaves their ratio unchanged: the two-state RNA melting model has no total-concentration term. By contrast, consider self-complementary RNA duplex melting, with association $2S\rightleftharpoons D$. Let $c_T=[S]+2[D]$ count strands and let $c^{\circ }$ denote the reference molar concentration of the standard state. The dimensionless association equilibrium constant isAt the midpoint defined by half the strands belonging to duplexes, $[D]=c_T/4$ and $[S]=c_T/2$, giving $K_a(T_m)=c^{\circ }/c_T$. This factor follows from strand counting; a different reaction or midpoint convention requires its own derivation.

An original ideal calculation assumes constant association enthalpy $\Delta H_a=-200\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ and a midpoint $330\mathrm{K}$ at $1\mu \mathrm{M}$. Increasing strand molar concentration tenfold givesor approximately $340.8\mathrm{K}$. Here $R$ is the molar gas constant. Greater strand availability favors association, so more heating is needed to reach the same strand-based midpoint.

Experimentally, one records RNA optical melting curves at several strand molar concentrations, with matched solvent conditions. A tetraloop study found concentration-independent transitions over approximately a 50-fold range, supporting intramolecular stem–loop formation.^[ref] This test is conditional on reversibility: a kinetically trapped association can fail to show the expected equilibrium concentration response.

Self-complementary RNA duplex melting dissociates a duplex formed by two copies of the same RNA strand into separate strands. Self-complementarity means that the strand's reverse complement can match its own nucleotide sequence. The reaction has molecularity two in the association direction, unlike intramolecular RNA hairpin melting.

Use the ideal association model $2S\rightleftharpoons D$. Let total strand molar concentration be $c_T=[S]+2[D]$ and paired-strand fraction be $f=2[D]/c_T$. With reference molar concentration $c^{\circ }$, the dimensionless equilibrium constant isSubstituting $[S]=c_T(1-f)$ and $[D]=f{c}_T/2$ givesThe final expression is the physical solution for a self-complementary RNA duplex population; the algebraically equivalent subtractive form loses numerical precision at small $\alpha$.

At $\alpha =1$, half the strands are paired. For an original $c_T=10\mu \mathrm{M}$ example, this means $[D]=2.5\mu \mathrm{M}$ and $[S]=5\mu \mathrm{M}$. Duplexes are only one-third of the independently diffusing molecular objects, $[D]/([D]+[S])=1/3$. Half the strands is not half the objects. At fixed temperature and $K_a$, increasing $c_T$ tenfold makes $\alpha =10$ and $f=0.8$.

Heating changes $K_a$ and thus the population. At the strand-based midpoint, $K_a(T_m)=c^{\circ }/c_T$, which explains the concentration dependence of RNA melting. A fit should state whether its constant describes association or dissociation and whether concentration counts strands or duplexes.

Real sequences can also form competing hairpins. A primary tetraloop study explicitly compared predicted hairpin and duplex equilibria when designing its optical-melting sequences.^[ref] The equations above assume only separated strands and duplexes; they do not automatically cover competing folds or RNA melting hysteresis.

A self-complementary RNA duplex population describes how identical RNA strands are distributed between free strands and duplexes under specified conditions. A fraction of strands in duplexes differs from a fraction of molecular objects that are duplexes. The distinction matters when comparing chemical mass balances with observed signals.

For the ideal equilibrium $2S\rightleftharpoons D$, let $c_T=[S]+2[D]$ and $f=2[D]/c_T$. With dimensionless association constant $K_a$ and reference molar concentration $c^{\circ }$, define $\alpha =K_a{c}_T/c^{\circ }$. The strand balance givesThis is the population model underlying ideal self-complementary RNA duplex melting. It assumes separated strands and duplexes are the only relevant species and that chemical equilibrium is reached.

The concentration regimes are illuminating. For $\alpha \ll 1$, $f\approx 2\alpha$; for $\alpha \gg 1$, $1-f\approx 1/\sqrt{2\alpha }$. An original calculation gives $f\approx 0.0192$ at $\alpha =0.01$ and $f\approx 0.9317$ at $\alpha =100$. Even a large association parameter leaves some free strands in this equilibrium model.

An original finite-count illustration has 100 strands,80 of which belong to 40 duplexes, with 20 free strands left. The paired-strand fraction is 0.8, but duplexes are $40/(40+20)=2/3$ of the 60 molecular objects. A detector responding equally to each object would not give the same signal as a detector responding equally per strand. RNA optical melting needs a model of the folded and unfolded optical responses to infer populations.

Differentiating the equilibrium relation givesAt $f=0.5$, a small 10% relative change in $\alpha$ changes $f$ by approximately 0.0167. Population is therefore not generally proportional to concentration.

Competing RNA hairpins require an expanded species balance. A primary tetraloop study evaluated hairpin and duplex formation explicitly when designing sequences for melting experiments.^[ref] Ignoring a populated competitor can produce a precise solution to the wrong balance equation.