= Self-complementary RNA duplex population
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 $\alpha=K_ac_T/c^\circ$. The strand balance gives
$$
\alpha=\frac{f}{2(1-f)^2},\qquad
f=\frac{4\alpha}{4\alpha+1+\sqrt{8\alpha+1}}.
$$
This 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 $\alpha\ll1$, $f\approx2\alpha$; for $\alpha\gg1$, $1-f\approx1/\sqrt{2\alpha}$. An original calculation gives $f\approx0.0192$ at $\alpha=0.01$ and $f\approx0.9317$ 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 gives
$$
\frac{df}{d\ln\alpha}=\frac{f(1-f)}{1+f}.
$$
At $f=0.5$, 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.<Thermodynamic characterization of naturally occurring RNA tetraloops by Justin P. Sheehy and others>[Materials and Methods, Design of sequences for optical melting studies.]{ref} Ignoring a populated competitor can produce a precise solution to the wrong balance equation.
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