RNA Secondary Structures Regulate Adsorption of Fragments onto Flat Substrates

RNA is a functionally rich molecule with multilevel, hierarchical structures whose role in the adsorption to molecular substrates is only beginning to be elucidated. Here, we introduce a multiscale simulation approach that combines a tractable coarse-grained RNA structural model with an interaction potential of a structureless flat adsorbing substrate. Within this approach, we study the specific role of stem-hairpin and multibranch RNA secondary structure motifs on its adsorption phenomenology. Our findings identify a dual regime of adsorption for short RNA fragments with and without the secondary structure and underline the adsorption efficiency in both cases as a function of the surface interaction strength. The observed behavior results from an interplay between the number of contacts formed at the surface and the conformational entropy of the RNA molecule. The adsorption phenomenology of RNA seems to persist also for much longer RNAs as qualitatively observed by comparing the trends of our simulations with a theoretical approach based on an ideal semiflexible polymer chain.

where z is the distance between the phosphate bead and the substrate. The interaction range is on the order of σ = 0.1 nm, which is substantially shorter than in the Debye-Hückel case, where it is λ D = 1 nm. The strength of the attraction was adjusted by the parameter surf , which ranged from 6.22 to 8.89 k B T in order to observe the adsorbed regime for both noss and wiss scenarios. Additional details concerning the length and Umbrella Sampling implementation of the simulations are in the Simulation Parameters section of the Supplementary Information. Figure S1: PMF curves for RNA fragments S1-noss (left panel) and S1-wiss (right panel) for RNA-substrate interaction using the Mie potential.
PMF curves for RNA fragment S1 in both noss and wiss forms are shown in Figure S1, and the minimum values of the PMF are depicted in Figure S2, displaying the same trend as in the case of the Debye-Hückel potential, analyzed in the main text. In Figure S2, we identify the same two adsorption regimes as in the main text, the first for surf < 7.25, where the fragment with a secondary structure adsorbs more strongly, and the second regime for stronger adsorption strengths ( surf >7.25), in which the unstructured molecule shows a more favorable adsorption. These results are comparable to those shown in Figure 4 and discussed in the main part of the article.
S2 Figure S2: Minimum of PMF for RNA fragment S1 using the Mie potential for RNAsubstrate interaction.
2 Monomer distributions Figure S3 shows the monomer distribution for RNA fragments S1-wiss and S1-noss, considering only those conformations where the molecule's center of mass is at a distance equal to or less than d m , where d m is the point at which PMF has half of the value of its minimum.
For RNA fragments S2 and S3, histograms showing their monomer distributions are shown in Figure S4.

S5
In Figure S6, we show the number of contacts for RNA fragments S2 and S3. The blue (S2-noss) and red (S2-wiss) curves indicate a similar behavior of the number of contacts with adsorption strength as shown in Figure 4B in the main text for RNA fragment S1.
Additionally, the number of contacts for fragment S3-wiss is expected to be equal to or greater than for S2-wiss. Figure S5: Number of contacts of RNA fragments S2 and S3 on the adsorbing surface interacting via the Debye-Hückel potential.

Radius of Gyration components
Considering that the number of coarse-grained particles is N and that the attractive surface is perpendicular to the z direction, the definitions of the normal R 2 g⊥ and parallel R 2 g contributions to the radius of gyration are given by where R CM = (x CM , y CM , z CM ) is the position of the center of mass. Tables S1 and S2 list the components of the radius of gyration for RNA fragment S1 in both its structured and unstructured variant, while Tables S3, S4, and S5 contain the components of the radius of gyration for S2-noss, S2-wiss, and S3-wiss RNA fragments, respectively.  The Collective Variable (CV) restraint consisted on a harmonic potential whose minimum was situated at a distance d 0 , which ranged between 0.5 nm and 5.5 nm for S1-wiss and between 0.5 nm and 6.5 nm for S1-noss, with a spacing of 0.5 nm. The values of ε surf /k B T were first determined in SPQR reduced units and further converted (as shown in the Simulation Parameters section). For systems S2 and S3, the CV restraints were located between 0.5 nm and 8 nm, with a spacing of 0.5 nm, and the surface strength was determined for four values as in the case of fragment S1. For each Umbrella Sampling simulation restraint, 24 independent simulations were performed.
The PMFs were shifted to zero at the positions where their centered derivatives were consistent with zero within the error bar, which is between 4.5 and 5 nm for S1-wiss and between 5.5 and 6 nm for S1-noss. For fragments S2 and S3, this domain was set between 7 and 8 nm. For the rest of the averages, the same weights were used on a sampled subspace of conformations. The average radius of gyration was calculated on a subset of structures S8 with a phosphate group at a distance from the surface below 0.37 nm, approximately one phosphate radius away from the surface potential minimum, as defined within the model.
In this analysis, the calculation of the number of contacts considered only snapshots whose value of d was below d m , the distance at which the PMF reaches half of its minimum value. Figure S6 show the PMF of the RNA fragments S2 and S3 .

S9
We briefly describe the basics of the formalism that we have used to estimate the free energy per monomer of an adsorbed ideal polymer chain composed of N monomers and with a bond length l. In our case, bond length is taken as the average distance between two phosphate groups of an independent simulation of RNA fragment S1-noss in the absence of a surface, resulting in l = 0.61 nm. If the probability of finding the first monomer at position r 0 is P (r 0 ; 0), the probability of finding its N -th unit at a position r can be expressed as Here, G 0 (r, r 0 ; N ) is the polymer's Green function, which for an ideal chain satisfies the where u(r) is an external potential exerted on the monomers.
The Edwards equation can be solved by writing an Ansatz of the form 1 considering that G(r, r 0 ; 0) = δ(r − r 0 ). The values of n introduced on the right-hand side are the eigenvalues of the linear differential operator L E = − l 2 6 ∇ 2 + βu(r). In addition, the functions ψ n (r) form a complete and orthonormal basis for the space of solutions given by the properties of L E . For large N , the expression of G can be approximated by where 0 is the smallest eigenvalue.

S10
Several important quantities can be obtained from the solution of the previous equation in terms of its eigenfunctions and eigenvalues. The most relevant for our purposes is the partition function Z N , written as up to a multiplicative constant, from which the free energy can be obtained under the present assumptions as plus an additive constant independent of N . In addition, the monomer concentration at r is given by with the boundary condition ψ 0 (0) = ψ 0 (x → ∞) = 0. The free energy per monomer is where the first and second terms of the right-hand side can be interpreted as the surface energy and the entropy loss due to the adsorption process, respectively. It is worth pointing out that the surface energy term resembles the number of contacts as defined in the main text.

S11
To compare this result with our simulations, we assume u(x) = −u w exp(−x/λ). The repulsive part is taken into account via a boundary condition and is not explicitly included in the above expression. Therefore, for comparison, we interpret u w as the minimum value of the surface potential used in the simulations. The most general solution of the one-dimensional Edwards equation which is consistent with the boundary conditions is where J α is the Bessel function of the first kind and order α, C is a constant which can be obtained by normalizing the monomer concentration, and α = 2σ 6| 0 |/l. Additionally, the boundary conditions require that the order of the Bessel function has to be greater than zero, while the eigenvalue 0 must be determined numerically. The free energy per monomer, together with its energy and entropy contributions, are shown in Figure S7

S13
(r x /a, r y /b, r z /c), with a = b = 0.5 nm and c = 0.3 nm, is used to define the vectors G(r ij ) = (sin(γr x /r), sin(γr y /r), sin(γr z /r), 1 + cos(γr)) Θ(r c −r)/γ, where γ = π/r c , Θ is the Heaviside function andr c is a cutoff parameter. For a particular fragment (or stem) denoted by s and composed of N s nucleotides, the ERMSD can be written which allows us to define the restraining potential as where K ss is the corresponding harmonic spring constant.

Simulation Parameters
We choose the temperature to be T = 9u cg for all the simulations listed here, where u cg is the SPQR energy unit, in order to obtain a better resemblance of the distribution functions sampled by the backbone potentials, originally obtained from an X-ray structure set. 4 This choice is a technicality that does not affect the results in practice because all the bonded interactions are essentially unaffected by this parameter, and all non-bonded interactions (apart from the excluded volume) are incorporated through the secondary structure restraint.
Thus, the simulated scenarios are below the melting temperature of the structured fragments, where unstructured chains of the same length can coexist.
In addition, both the glycosidic bond angle state and the sugar pucker were kept fixed S14 to their anti and C3 endo conformations during the calculations to avoid the introduction of additional parameters. In these units, the chosen values of surf described in the main text have values of 4u cg , 6u cg , 8u cg , 10u cg , 12u cg , 14u cg , and 16u cg . This range of values of surf allows us to clearly distinguish the two adsorption regimes described in the main text. Secondary structure was fixed with a harmonic restraint on the ERMSD, with an dimensionless cutoffr = 100 and a harmonic spring of K ss = 50 cg .
The RNA fragments S1, S2, and S3 with secondary structure were constructed imposing the secondary structure conditions until the contacts were formed, relaxed with different random seeds and integrated for 10 6 Monte Carlo (MC) sweeps for the fragment S1 and 10 8 MC sweeps for the fragments S2 and S3. Each MC sweep consists of N MC steps, where N is the number of nucleotides. From here, the unstructured fragments were generated for each initial condition, running a simulation at constant temperature for 10 6 MC sweeps.
It was noticed that starting from a helical conformation, as a single strand extracted from a template stem in the A-form, gave a consistent end-to-end distance after the relaxation procedure. For systems S2 and S3, the procedure was the same, although the equilibration simulations for decorrelating the structured fragments and generating the unstructured initial conditions consisted of 10 8 MC sweeps. After this, the surface was positioned perpendicular to the z direction, 0.1 nm from the closest nucleotide. The simulations were performed for 1.5 × 10 8 MC sweeps, and configurations were saved every 50000 steps, while the first 750 saved configurations were discarded. For fragments S2 and S3, the runs consisted of 3 × 10 8 MC sweeps, and configurations were saved every 10000 steps while the first 750 saved configurations were again discarded.

S18
For the simulations with the Mie potential, the harmonic restraints of the Umbrella simulations were situated at d 0 = 0.3 nm and from 0.5 to 4 nm, with steps of 0.5 nm and k u = 5.56 nm 2 /k B T . As in the previous system, in some cases, additional simulations using different Umbrella restraints were performed or omitted because of their long autocorrelation times, which are described in Table S7. For each setup, 24 independent runs were analyzed, consisting of 150 × 10 6 MC sweeps, and configurations were saved every 50000 sweeps. The first 750 saved configurations were discarded. Table S7: Details of Umbrella Sampling simulations for fragments S1-wiss and S1-noss interacting with the Mie potential.