### Kinetic Models Describe the Folding Reaction  
We consider the process by which an RNA molecule adopts a single, native conformation as illustrated in Figure 1A. RNA secondary structure is highly stable and is formed in several microseconds [1; 15; 16; 17]. The folding process we describe here involves the conformational rearrangement of the secondary structure elements (helices and junctions) that is induced by the addition of a counter-ion (in general Magnesium salts [17]) which leads to a native, and therefore active RNA molecule. This process is the rate-limiting step in RNA folding, due to the appearance of multiple, long-lived intermediates along the folding pathways [7]. We describe the folding process by a kinetic model, such as that illustrated in Figure 1A. The RNA begins in the unfolded state (\(U\)) and folds either directly or through intermediates (\(I\)) to the final, active state (\(F\)). Mathematically, the kinetic model can be written as a series of ordinary differential equations, with rate constants (\(k\) and \(r\)) describing the rate of transition between states in inverse seconds. The rate of change of the unfolded state, U(t) is therefore described by  
\[\frac{dU(t)}{dt}=-k_{U1}U(t)-k_{U2}U(t)\ldots-k_{UF}U(t)+r_{U1}I1(t)+r_{U2}I2( t)\ldots+r_{UF}F(t) \tag{1}\]  
In this case, \(k>>r\) such that a majority of the molecules ultimately reach the final folded state (\(F\)) and the unfolded state (\(U\)) is almost completely unpopulated when steady state (\(t=\infty\)) conditions are Figure 1: Basic premise for describing an RNA folding reaction with a kinetic model. A.) Unfolded RNA (\(U\) state) has only secondary structure elements formed (secondary structure diagram shown in the left hand circle) while the folded state (\(F\)) has the full complement of tertiary interactions which allow it to adopt a unique three-dimensional structure (shown in the right hand circle). The folding reaction often takes several hours to go to completion due to multiple intermediates \(I\) that populate the pathways to the \(F\) state. Each transition between state curves is indicated with an arrow and has a rate constant associated with it. For clarity, we only show the forward transitions in this diagram as the forward rates are generally much larger than the reverse rates. B.) Resulting state curves \(\vec{x}(t)\) for \(U\) (orange), \(I1\) (magenta), \(I2\) (purple) and \(F\) (black) that describe the relative fraction of each species as a function of time for the folding of the L-21 _T. thermophila_ group I intron in the presence 10 mM MgCl\({}_{2}\)[9]. These curves are obtained from Equation 5. C.) -OH footprinting curves (\(\vec{C}_{P}(t)\)) for folding of the _T. thermophila_ group I intron in the presence of 10 mM MgCl\({}_{2}\). The green curve corresponds to the P4P6 subdomain, red to the periphery, and blue to the catalytic core of the molecule; a corresponding color scheme is used to illustrate these subdomains for the structures shown in A. We assume that the progress curves are a result of different fractions of the unfolded state and the fully folded state. achieved. If we now consider a vector of the individual state curves, \(\vec{x}(t)\), defined as  
\[\vec{x}(t)=\left( \begin{array}{c}U(t)\\ I1(t)\\ I2(t)\\ \vdots\\ F(t)\end{array} \right) \tag{2}\]  
and further define matrices **K** and **D**, as shown  
\[\textbf{K}=\left[ \begin{array}{ccccc}0&k_{U1}&k_{U2}&\ldots&k_{UF}\\ r_{U1}&0&k_{12}&\ldots&k_{1F}\\ r_{U2}&r_{12}&0&\ldots&k_{2F}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ r_{UF}&r_{1F}&r_{2F}&\ldots&0\end{array} \right] \tag{3}\]  
\[\textbf{D}_{ij}=\left\{ \begin{array}{l}i\
eq j;\ \ \ \ \ \ \ \ \ \textbf{K}_{ji}\\ i=j;\ \ -\sum_{i=1}^{n}\textbf{K}_{ji}\end{array} \right. \tag{4}\]  
then the kinetic model illustrated in Figure 1A is written as  
\[\frac{d\vec{x}(t)}{dt}=\textbf{D}\vec{x}(t). \tag{5}\]  
Numerical integration of Equation 5 for a given **K** matrix (set of rate constants) yields the state curves (\(\vec{x}(t)\)) illustrated in Figure 1B.