Mathematica 12

To characterize the structural features of the systems under study, we carried out Born–Oppenheimer ab initio molecular dynamics simulations for the bis(3-hydroxy-4-pyronato) oxovanadium(IV) complex, VO(3hp)₂, and the bis(maltolato) zinc(II) complex, Zn(mal)₂, in vacuum and in water clusters made of 25 molecules using the CP2K program [66 (link)]. We performed NVT (constant volume, constant temperature, using a Nosé thermostat [67 ]) simulations in cubic boxes with side-lengths of about 17 Å (the length varied ±2 Å from system to system) under periodic boundary conditions. We employed the generalized gradient approximation and the Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional [68 (link),69 (link)]. The Grimme D3 approach is taken to account for dispersive interactions [70 (link)] and the dzvp-molpot double-ζ polarization basis [71 ] is used for the Geodecker–Teter–Hutter pseudopotentials [72 ]. A time-step of 0.5 fs was employed. In each case, we performed a thermalization phase under a velocity rescaling regime [73 (link)]. Typically, thermalization in the prepared systems was reached within 300 fs. Next, we conducted NVT sampling of 1 ps. The target accuracy for the self-consistent field convergence was 10⁻⁶ hartree. The cut-off and the relative cut-off of the grid level were set to 400 and 100 Rydberg, respectively. The energy convergence threshold was set to 10⁻¹². We conducted data analysis as well as the preparation and training of the convolutional Neural Net using Mathematica 12, Wolfram Research Inc., Champaign, Illinois, USA.

One serum, urine, and saliva sample from each sampling time point and each participant was processed and analyzed as described above. Maximum concentration (c_max) and time to reach maximum concentration (t_max) in serum were determined for each participant. Elimination constant (k_e) and elimination half-life (t_1/2) of the analytes in serum were calculated using a non-compartmental analysis in the PKanalix 2024 software (Lixoft, Antony, France) applying Eq. (1).
For compartmental analysis the measured data were fitted in Wolfram Mathematica 12 (Wolfram Research, Champaign, USA) using the two-compartment model with first-order absorption.
Creatinine concentrations used to normalize urine concentrations of the analytes were determined in an external, accredited laboratory. Serum concentrations as well as urine concentrations normalized for creatinine were plotted against time using IBM SPSS Statistics 29.0 (IBM, Armonk, New York, USA).

Under our experimental conditions, the confocal volume is most often devoid of fluorescent molecules, with an average occupancy of less than 0.01 molecules. Nevertheless, individual fluorescently labeled molecules will inevitably diffuse through the confocal volume, where they will be excited by the two alternating lasers, resulting in a transient burst of fluorescent photons (Fig. 2B). The detection records of the photons emitted from these freely diffusing fluorescently labeled RNA molecules were analyzed using Mathematica 12.0 (Wolfram Alpha) in conjunction with Fretica, a C++ based MATHLINK module for the analysis of time-correlated single-photon-counting single-molecule FRET data (40 ). The data analysis workflow employed during this research is as follows: first, time-gating was used to determine which laser (either 515 or 642 nm) was active for every detected photon in each of the four streams. Then, photons were assigned to 500 μs time bins based on their absolute arrival time (Fig. 2C). Time bins with a total photon count rate (^Totn = ¹n₅₁₅ + ¹n₆₄₂ + ²n₅₁₅ + ²n₆₄₂ + ³n₅₁₅ + ³n₆₄₂ + ⁴n₅₁₅ + ⁴n₆₄₂) of less than 20 photons per bin were used to calculate the average background photon count rate for each of the four streams during either 515 or 642 nm excitation. Then, corrected photon count rates (N) were determined for all time bins by accounting for background, spectral cross talk, direct excitation of the acceptor, and nonidentical excitation and detection efficiencies of the donor and acceptor fluorophores. Those time bins with a corrected total photon count rate (^TotN = ¹N₅₁₅ + ¹N₆₄₂ + ²N₅₁₅ + ²N₆₄₂ + ³N₅₁₅ + ³N₆₄₂ + ⁴N₅₁₅ + ⁴N₆₄₂) of more than 20 photons per bin were considered bursts of fluorescence arising from single molecules diffusing through the confocal volume (41 (link)). Corrected photon count rates associated with the acceptor and donor fluorophores (^AccN = ¹N + ³N and ^DonN = ²N + ⁴N, where ^TotN =^AccN + ^DonN) during 515 and 642 nm excitation were used to calculate values for both the FRET efficiency (E) and fluorescence stoichiometry (S) using Eq. 1 and Eq. 2, respectively (Fig. 2D). $E={}^{Acc}N_{515}/\left({}^{Acc}N_{515}+{}^{Don}N_{515}\right).$ $S={}^{Tot}N_{515}/\left({}^{Tot}N_{642}+{}^{Tot}N_{515}\right).$
The fluorescence stoichiometry (S) of a burst arising from a molecule containing only donor or acceptor fluorophores will yield values near S = 1 and S = 0, respectively. Therefore, values of fluorescence stoichiometry are used to restrict our analysis and interpretation of FRET efficiencies to only those bursts arising from molecules containing active donor and acceptor fluorophores (i.e., 0.25 < S < 0.75). This allows us to effectively filter out unwanted contributions from any potential donor-only or acceptor-only molecules that, for example, may not have been removed during the HPLC purification of the RNA constructs.
The FRET efficiency (E) values from bursts with ^TotN > 50 were then compiled into histograms (Fig. 2D), in which the widths of the resulting distributions were largely limited by shot noise (Fig. S1). Histograms were then fitted using Gaussian distributions to determine the mean FRET efficiency, ⟨E⟩, and fractional abundance, Θ, of the folded and unfolded subpopulations. Based on several repeated measurements under identical conditions, typical experimental uncertainties associated with ⟨E⟩ and Θ are ±0.02 and ±0.03, respectively. The fractional abundance of each subpopulation was used to calculate the equilibrium constant for folding (K_fold = Θ_f/Θ_u) and thus the standard state Gibbs free energy difference (ΔG°_fold = −RT ln K_fold, where R is the gas constant) between the two subpopulations at T = 294.2 K. For the ease of data interpretation, all ΔG°_fold values in apolar solvent conditions were referenced to aqueous conditions, i.e., ΔΔG°_fold = ΔG°_fold (mixture) − ΔG°_fold (H₂O).

Scattering patterns of CNCs suspensions were collected using SAXSLAB GANESHA 300-XL Xenocs, Grenoble, France. CuΚα radiation was generated by a Genix 3D Cu-source with an integrated monochromator, 3-pinhole collimation, and a two-dimensional Pilatus 300 K detector. The scattering intensity I(q) was recorded in the λ interval of 0.007 < q < 0.25 Å⁻¹ (corresponding to the length scale of 25–900 Å), where the scattering vector is defined as q = (4π/λ) sin θ, with 2θ and λ being the scattering angle and wavelength, respectively. The measurements were performed under vacuum at ambient temperature. The suspensions were sealed in thin-walled quartz capillaries about 1.5 mm in diameter and with 0.01 mm wall thickness. The scattering curves were corrected for counting time and sample absorption. The 2D SAXS patterns were azimuthally averaged to produce one-dimensional intensity profiles, I vs. q, using the two-dimensional data reduction program SAXSGUI. The scattering spectra of the solvent were subtracted from the corresponding solution data using Igor Pro 9 from WaveMetrix Portland Oragon for analysis of small-angle scattering data [36 (link)]. Data analysis was based on fitting the scattering curve to an appropriate model provided by Mao et al. [37 (link)] using Wolfram Mathematica 12.3 software, Champaign, IL, USA.