This manuscript presents a mathematical model and a description of a numerical simulation of the electrochemical processes surrounding transmuting DNA origami nanostructures within a capacitance-signaling biosensor. A modified Poisson-Boltzmann equation is employed to predict the electric potential about a single DNA origami nanostructure, represented by variability of permittivity and background charge within the domain. Differential capacitance is used to determine the capacitance signal from model variables. The solution to the model, as well as the capacitance, is numerically approximated using finite differencing. Nonsmooth and smooth nanostructure treatments are motivated and analyzed, suggesting that while the nonsmooth treatment introduces large numerical error and loses an order of accuracy, the difference in capacitance between the two treatments is far less significant, though still apparent. A parameter study informs numerical refinement choices and demonstrates the variability in solution caused by the modification of ions species in solution as well as maximum ion concentrations. Preliminary observations on device sensitivity are also provided, suggesting that nanostructures placed away from the bottom surface of the device can cause a reduction in sensitivity and some nanostructure designs are more advantageous for capacitance signaling.
Citation: Allison M. Carson, Wes Caldwell, Ryan M. Evans, Anthony J. Kearsley. Mathematical model of electrochemical signaling from transmuting DNA origami nanostructures within biosensors[J]. Mathematical Biosciences and Engineering, 2026, 23(11): 3211-3240. doi: 10.3934/mbe.2026111
This manuscript presents a mathematical model and a description of a numerical simulation of the electrochemical processes surrounding transmuting DNA origami nanostructures within a capacitance-signaling biosensor. A modified Poisson-Boltzmann equation is employed to predict the electric potential about a single DNA origami nanostructure, represented by variability of permittivity and background charge within the domain. Differential capacitance is used to determine the capacitance signal from model variables. The solution to the model, as well as the capacitance, is numerically approximated using finite differencing. Nonsmooth and smooth nanostructure treatments are motivated and analyzed, suggesting that while the nonsmooth treatment introduces large numerical error and loses an order of accuracy, the difference in capacitance between the two treatments is far less significant, though still apparent. A parameter study informs numerical refinement choices and demonstrates the variability in solution caused by the modification of ions species in solution as well as maximum ion concentrations. Preliminary observations on device sensitivity are also provided, suggesting that nanostructures placed away from the bottom surface of the device can cause a reduction in sensitivity and some nanostructure designs are more advantageous for capacitance signaling.
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Figures(19) / Tables(5)
Allison M. Carson, Wes Caldwell, Ryan M. Evans, Anthony J. Kearsley. Mathematical model of electrochemical signaling from transmuting DNA origami nanostructures within biosensors[J]. Mathematical Biosciences and Engineering, 2026, 23(11): 3211-3240. doi: 10.3934/mbe.2026111
An artistic rendition of a contorting DNA origami nanostructure. Each gray rod represents a double strand of DNA, with two yellow connected strands representing aptamers that limit how wide the nanostructure's upper arm may open. The analyte of interest, if present within the patient's sample, would replace the binding of one of the aptamers, untethering the two such that the upper arm is able to open to a wider angle. The displacement of the DNA alters the established electric field, thus causing device signaling
Schematic of modeled domain (not to scale)
Smoothing regions for an example rectangle. The rectangle boundaries are shown in red, yellow, blue, and green, which outline $ \Omega_{DNA} $. The associated smoothing region of each rectangle segment is highlighted in the corresponding lighter color (e.g. the light red region corresponds to the red segment). The smoothing regions are separated along diagonal axes highlighted by the black dotted lines
Example numerical solutions highlighting solution variability. Computations took $ N_x = 2560 $, 12.5 mM MgCl$ _2 $, $ \theta = 0 $, and the smoothing treatment of the nanostructure
Capacitance curves from CV sweeps calculated with (w/) and without (w/o) the DNA origami nanostructure. Three cases of ionic strength are plotted, given as molarity of NaCl in the legend. Computations took $ N_x = 2560 $, 12.5 mM MgCl$ _2 $, $ \theta = 0 $, and the smoothing treatment of the nanostructure
Distribution of $ e_{\phi} $ (left) and its discretization order-of-accuracy (right) computed across applied voltage and ionic strength, listed as molarity of NaCl in the legend. Results for the smooth nanostructure treatment (S) are plotted in solid lines and the nonsmooth (NS) dashed lines
Estimated error $ e_{\phi} $ associated with $ \tilde \phi_0 = .4 $ V and 50 mM of NaCl for the nonsmooth (left) and smooth (right) nanostructure treatments
Capacitance calculated with the nonsmooth (NS) and smooth (S) nanostructure treatments as a function of applied voltage and ionic strength, given as molarity of NaCl in the legend
Magnitude of the relative difference between capacitance curves from CV sweeps with the nonsmooth $ \left(\tilde c_{T, h}^{(NS)}\right) $ and smooth $ \left(\tilde c_{T, h}^{(S)}\right) $ nanostructure treatments as the computational grid is refined. Three cases of ionic strength are plotted, given as molarity of NaCl in each title
The top left plot shows the estimated numerical error $ e_{\phi} $ across applied voltage and ionic strength. Computations taking $ N_x = 1280, 2560 $ were used in the estimation. All other plots show capacitance curves from CV sweeps with decreasing $ \tilde l $. Three cases of ionic strength are plotted, given as molarity of NaCl in each title. Capacitance curves associated with the nonsmooth nanostructure are plotted in black for comparison
Plot of average time elapsed over five computations for each capacitance calculation across applied voltage and ionic strength, given as molarity of NaCl in the legend. The nonsmooth (NS) nanostructure treatment is plotted with the dashed lines and the smooth (S) treatment is plotted with the solid lines. Computations took $ N_x = 2560 $, $ tol = 10^{-8} $, and $ y_b = 0.5 $, and were run in serial using an Intel Xeon 3.6GHz processor with the gcc compiler and -O3 optimization flag
Distribution of $ e_c $ according to $ h $ (left) and $ \delta\tilde\phi_0 $ (right) computed across applied voltage and ionic strength, listed as molarity of NaCl in the legend. Absolute errors may be thought of the relative error multiplied by a factor of $ 10^{-6} $
Capacitance curves from CV sweeps calculated with no, half-concentration, and full-concentration of PBS. Three cases of ionic strength are plotted, given as molarity of NaCl in each plot title
Capacitance curves from CV sweeps calculated with different values of $ \gamma $. Three cases of ionic strength are plotted, given as molarity of NaCl in each plot title
Difference in capacitance between a computation of stilt height $ \tilde S_H $ and a computation without the DNA origami nanostructure. Values in the legend are provided in nanometers
Estimated error in $ \Delta \tilde c_{T, h}^{(\tilde S_H)} $ for the case of $ \tilde S_H = 1 $ nm and 500 mM of NaCl (left) as well as $ \tilde S_H = 1.8 $ nm and 1 M of NaCl (right). Error for a coarse grid, computed with $ N_x = 2000, 2500 $ and a fine grid, $ N_x = 2500, 3000 $, are shown
The three candidate nanostructure designs, shown at $ \theta = \frac{\pi}{4} $
LEFT: Example capacitance signal of each nanostructure design over the range of nanostructure angle. RIGHT: Capacitance signal of the single arm design as the computational grid is refined
Representation of nanostructure shape signals over CV sweeps with varying ionic strength, which is given as molarity of NaCl in the legend
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