Far-Field Electrostatic Signatures of Macromolecular 3D Conformation

In solution as in vacuum, the electrostatic field distribution in the vicinity of a charged object carries information on its three-dimensional geometry. We report on an experimental study exploring the effect of molecular shape on long-range electrostatic interactions in solution. Working with DNA nanostructures carrying approximately equal amounts of total charge but each in a different three-dimensional conformation, we demonstrate that the geometry of the distribution of charge in a molecule has substantial impact on its electrical interactions. For instance, a tetrahedral structure, which is the most compact distribution of charge we tested, can create a far-field effect that is effectively identical to that of a rod-shaped molecule carrying half the amount of total structural charge. Our experiments demonstrate that escape-time electrometry (ETe) furnishes a rapid and facile method to screen and identify 3D conformations of charged biomolecules or molecular complexes in solution.

Oligonucleotide synthesis. DNA oligonucleotides were ordered from Integrated DNA Technologies, Inc (IDT), unpurified except for those with the 5'Atto-532 fluorescent label (HPLC purified). Strands were resuspended at a concentration of 100µM in TE Buffer (10mM Tris + 1mM EDTA).
Assembly. Annealing protocols for each nanostructure are specified separately below.
Gel purification. Gels (run as above) were imaged in the Atto532 fluorescence channel (532nm laser and 570nm bandpass filter). The band corresponding to a fully-assembled nanostructure was cut out and nanostructures eluted by adding 200 µL TE Buffer with 5 mM MgCl2 (tetrahedron, two-helix bundle, 120bp DNA helix) or 12.5 mM MgCl2 (square tile), freezing and thawing twice and collecting the supernatant. Tetrahedron and square tile samples were further concentrated by ethanol precipitation 1 and resuspended in TE Buffer alone (square tile) or in TE Buffer with 5 mM MgCl2 (tetrahedron).
Quantification. Purified sample concentrations were estimated on the basis of the intensity of emission from the Atto532 fluorophore. Fluorescence intensities were recorded using a CLARIOstar Plus (BMG Labtech) Microplate reader (excitation: 520nm, emission: 570nm). A calibration curve was obtained using the 5'Atto-532 labeled strand T_s1 at known Assembly. Labeled strand (S2) at 200 nM and unlabeled strands ( Figure S3, B_s1, B_s2, B_s3) at 300nM were combined in TE Buffer with 12.5 mM MgCl. The mixture was heated to 95°C, held for one minute, then cooled at 1°C per minute to 22°C.

1.5: 120bp DNA helix synthesis
Design. The 120bp helix is assembled from four strands of DNA to circumvent the requirement to synthesize an Atto532-labeled oligonucleotide of length 120 nucleotides ( Figure S4a).
Strands S1 and S2 from the 2-helix bundle are re-used in this structure.
Assembly. Labeled strand (S1) at 200nM and unlabeled strands (S2, Ds_s1, Ds_s2) at 300 nM were combined in TE Buffer with 12.5 mM MgCl. The mixture was heated to 95°C, held for one minute, then cooled at 1°C per minute to 22°C.

1.6: Determining the structural charge of DNA nanostructures
The structural charge, , is determined for each of the DNA nanostructures as follows.
Beyond summing the number of bases, (each associated with a charge of -1e due to a phosphate ester group), we also consider the number of strands, , in each structure, and whether their 5' termini are phosphorylated or not. In the case of all the nanostructures, ( − 1) strands had hydroxylated 5' termini, while the dye-carrying " th strand" did carry a 5' phosphorylation to which the dye is chemically coupled.    each lane contains the unpurified products of assembly of one more strand than the previous lane. Lane 1: S1; Lane 2: S1 + B_s2; Lane 3: S1 + B_s2 + S2; Lane 4: S1 + B_s2 + S2 + B_s1; Lane 5: S1 + B_s2 + S2 + B_s1 + B_s3 Figure S4. Design & assembly of the 120bp DNA helix. (a) 120 bp DNA Helix (b) Formation gel: each lane contains the unpurified products of assembly of one more strand than the previous lane. Lane 1: S1; Lane 2: S1 + Ds_s2; Lane 3: S1 + Ds_s2 + S2; Lane 4: S1 + Ds_s2 + S2+ Ds_s1.   tuned by adding NaCl. We parametrize experimental conditions using the dimensionless product, ℎ, of inverse electrostatic screening length, (inverse Debye length), and h, which denotes half the height of the slit. is obtained from electrical conductivity measurements of the measured aqueous suspension in the device, after each ETe measurement (HORIBA LAQUAtwin), as described previously 9 . The slit height, 2h, is characterized using atomic force microscopy (Asylum Research); the device fabrication procedure is similar to that in previous work 10 .
For a single measurement of a nanostructure species at a specific ionic strength, molecules are loaded into nanoslits housing several arrays of traps using a vacuum suction system. Once loaded, the vacuum is turned off initiating free diffusion of the DNA nanostructures across the trap landscape. Fluorescent signals from individual nanostructures were recorded using a standard wide-field fluorescence microscope, imaged using 5ms exposure times at frame rates from 5 to 50 Hz for up to 1000 frames on a sCMOS camera (Prime 95B, Photometrics). More molecules were then loaded into the trap arrays, and the procedure repeated about 20 times per measurement. From these videos of DNA nanostructure diffusion in the electrostatic free energy landscape, molecular residence times, Δ , in each trap were extracted using customized video analysis software. The average escape time, , for each measurement was determined by fitting the probability density distribution, (Δ ), with a function the form (Δ ) = exp(− ) , as described in previous work 4,11,12 . Previous studies confirm that photobleaching does not affect the measurement of escape times in our experiment 9 .
2.2: Brownian dynamics (BD) simulations of escape process to convert the measured escape times, tesc, to well depth, , in the free energy landscape.
We converted measurements of escape time to potential well depth values, , as described previously 4,9,11,12 . As a general principle, when > 4 , we expect an exponential dependence of on of the form Here, is an effective molecular position relaxation time which is inversely related to the molecular diffusion coefficient, , i.e., ∝ . While equation S1 holds for large well depths and provides qualitative insight into the underlying dynamics, it does not strictly hold for well depths < 4 . In practice we do not know a priori the value of for a given molecular species. We therefore perform Brownian Dynamics (BD) simulations of the escape process in order to accurately relate to as described previously 9,12 . We simulate the thermal motion of a charged object in an electrostatic energy landscape, where the object is characterised by a friction coefficient given by 6 , while moving at velocity, , through a medium of dynamic viscosity, . The hydrodynamic radius, , of the molecular species is determined using two-focus Fluorescence Correlation Spectroscopy (2fFCS) to measure the molecular diffusion coefficient, which in turn may be expressed in terms of the radius of an equivalent sphere, (see section 2.3). The electrostatic potential landscape in the fluidic nanostructure is calculated by numerically solving the Poisson-Boltzmann (PB) equation, for our device geometry, subject to the appropriate boundary conditions, using COMSOL Multiphysics as described previously 9,12 . The electrostatic interaction energy of the particle at any point in the landscape, referenced to a free energy at infinite slit height, is given by = ( ), where ( ) is the electrical potential at . A periodic array of axially symmetric potential wells, constructed from the minimum free energy profile of a charged object traversing a single trap, is used for subsequent BD simulations (main text Figure 1c).
Simulated trajectories of particle motion in the landscape are used to generate image stacks similar to the experimental movies, as described previously 12 . Briefly, simulated movies are analysed in the same way as experimental movies in order to extract escape times (or average residence times of particles at trap locations) as a function of known input well depths, . While the generation of simulated movies fosters high precision measurements (<1% on ), movie generation is computationally expensive and not necessary for measurements with lower accuracy requirements (uncertainty around 5%). Here, we work directly with the spatial coordinates corresponding to the trajectory of particle motion in the landscape as described in detail previously 9 . We apply spatial entry and exit criteria to the simulated particle trajectory in order to define a trapping event. We point out that the molecular entry and exit criteria required in the trajectory-based analysis are initially determined by a tuning procedure which aims to recover the same escape time in both the simulated trajectory-based analysis and in the simulated movies. Here we first generate simulated movies of the escape process using the simulated molecular trajectories and the same signal-to-noise and brightness characteristics as the experiment. We then analyse the simulated movies in the same way as the experimental recordings. The entry and exit criteria in the trajectory-based analysis are fine-tuned until we obtain excellent agreement between average escape times from the particle trajectory and the simulated movie analyses. Thereafter we use just the simulated trajectory data, in conjunction with the entry and exit criteria, in order to theoretically determine escape time values that correspond to known input well depths, . By comparing experimentally measured escape times with theoretically expected escape times determined for a range of well depths, , for a particle of known , we determine the value of the unknown well depth in the experiment.
We see from equation S1 that the escape time depends exponentially on W and hence on the effective charge of the molecule, but only linearly on its size. Therefore our measured values do not depend sensitively on accurate knowledge of molecular size, and a reasonable estimate of will generally suffice. Note that as always remains unknown at the outset, we do not a priori use equation S1 to convert to . Once a set of simulations is performed for various value of , we generally find that an exponential function of the form shown in equation S1 may be fit in the high well depth regime. For the DNA nanostructures in the study, we find ≈ 1.9 ms, while for 60bp DNA, ≈ 1.4 ms, under the experimental sampling conditions. subject to the appropriate boundary conditions for the experimental system as described previously 14,15 (see section 3 for further detail). Here = ⁄ is the dimensionless local electrical potential, . We have previously shown that Δ is given by the product of an effective charge of the molecule in solution, , and the difference in midplane electrical potential, Δ , between the slit and nanoscale pocket regions 14 (main text Figure 1).
Denoting the electrical potential at the midpoint between the parallel-plate slit surfaces as , and recognizing that = 0 by design in the pocket region, we therefore have We emphasize that the relationship = (r), on which Eq. S4 is based, has been shown to hold rigorously in theoretical treatments such as Kjellander's "dressed ion" theory 16,17 .
Thus in order to obtain a measure of Δ and hence of from the inferred well depth, W, we first need to determine the fluctuation contribution, f, to the total well depth, .
The dominant contribution to the fluctuation free energy comes from axial fluctuation of the molecule in the z-dimension in the slit as described previously 4,14 . There is also a small contribution from a rotational term which we consider for non-spherically symmetric objects.
We determine = Δ + Δ by considering the differences in an 'excess' free energy due to translational (Δ ) and rotational (Δ ) fluctuations of the molecule between the two states of our parallel plate system. Δ accounts for the thermal motion of a molecule about the minimum-energy midplane of the system. As shown previously, we can write Δ in terms of the partition function, Z, of a point charge diffusing freely in z as follows 4 : represents the electrostatic free energy of the charged object located at any axial position, z, in the slit (referenced to the global minimum in interaction energy ( → ∞)).
Solving the equation S3 for a monovalent, binary and symmetric electrolyte, subject to constant charge boundary conditions, shows that the electrical potential at a location z in a parallel plate slit can be approximated as ( ) = cosh − .
Using equations S6 and S7, we evaluate Z for particle states 1 and 2 (the "slit" and "pocket" states, see Fig. S8a), with upper limits on the integral given by = 2ℎ or (2ℎ + ), dictated by the maximum parallel plate separation in the slit (2ℎ) or in the pocket region (2ℎ + ). Note that ( 2 ⁄ ) = is the electrical potential at the midplane of the slit.
Δ , in turn, accounts for the additional free energy due to rotational motion of a non-spherical particle. By approximating the DNA nanostructure as a line charge of length, , given by the longest dimension of the nanostructure, carrying a charge density = ⁄ , we can write Δ in terms of the partition function, Ω, of a freely rotating rod at an angle, , to the z-axis, as follows: Here we write the following expression for the local angular electrostatic energy, Δ In order to simplify the calculation of the rotational term, we have assumed that the charged object is confined to the midplane of the system, i.e. we only consider rotational contributions for the molecule located at = 2 ⁄ . The accuracy of this approximation, which permits

2.5: Calibration of the ETe measurement
We see from equation S4, that to make an accurate inference on from the inferred value of Δ we also require accurate knowledge of , the electrical potential at the midplane of the slit. depends on the effective surface electrical potential at the silica walls of the slit, as follows = 2 exp(− ℎ) (S11) Treating as constant for experiments performed over a narrow range of salt concentrations (1.1-5.5 mM) in a given device, and combining equations S4 and S11 yields ln = ln − ℎ (S12) We measured Δ for all the molecular species in our study under various conditions of salt concentration and slit height and plotted the measurements vs. a dimensionless system size, ℎ ( Figure S7b). Fits of the data to equation S12 yield values for the y-intercept given by 2 in each case. Knowing the value of the surface potential in the device would permit us to determine for each molecular species. Since an accurate value of is not available a priori we perform a calibration measurement using a well-characterized molecular species, whose geometry is well defined and whose effective charge, , is known from calculation 14 .
Here, we typically use a 60bp double-stranded DNA fragment carrying two fluorophore labels as a calibration molecule, for which we expect ≅ −45 for measurements performed at ≈1 mM monovalent salt, similar to previous work 12,15 . Corresponding measured effective charge values, denoted as , are displayed in Figure S7b. In general we find that the value of obtained for each device is stable over a timescale of several days and hence a single calibration run can be used for reliable measurements of effective charge, , of further species under the same or similar conditions.
We point out that in order to enable a direct quantitative comparison of electrostatic interactions for the various molecules in the study, we work with measured effective charge values, , rather than measured free energies, Δ . This is because is a property characteristic of the molecule and is robust to small variations in experimental conditions from one realization to the next. Δ on the other hand, which is proportional to , is very sensitive to small variations in slit height and salt concentration, embodied in a single parameter ℎ as shown in equation S12 and discussed in Figure S7. Division of the measured Δ value by yields a measured value of effective charge, , characteristic of the molecule of interest. Unlike and Δ , is robust to small changes in system size, ℎ, and permits direct comparison of measurements across independent realizations. Figure S7c displays individual measurements of Δ converted to measured effective charge values, , plotted vs. ℎ. Note that at = 1.5 mM NaCl e.g., the slight dependence of on salt concentration would imply ≈ −46.2 for 60bp DNA, which is about 3% larger in magnitude than the value at 1 mM NaCl (outlined in section 3.3). However, values for the DNA nanostructures are also expected to be similarly affected by salt concentration largely cancelling the slight influence of varying salt concentration in the final results ( Figure   S9). Measurements for all molecular species considered indeed show that values remain constant within measurement uncertainty over a wider range of salt concentrations of ~1-6 mM (Fig. 1e).

2.6: Sources of error in
In an ETe measurement that includes a calibration molecule, the uncertainty on , denoted as , , comes mainly from the statistical uncertainty of measuring N independent random events from a Poisson point process, and is given by   values denote errors on fits of (Δ ) vs. Δ data as shown in Figure 1 in the main text, while uncertainties in ℎ arise from accuracy limits on the measurement of slit height, ℎ (uncertainty ~1 nm), discussed previously 9 . Lines are guides to eye. (b) Electrostatic trap depth (Δ ) inferred from the escape time measurements in (a), as described in the text. Fits of the data to equation S12 yield a measured molecular effective charge, , with the silica surface potential, , calibrated for each device using 60bp double-stranded DNA (as described in the text). where the infinite rod model suggests ≈ 0.3. Overall, the comparison of our calculations for finite-length rods with a calculation for an infinite rod is not expected to be favourable for smaller rod lengths since end effects do begin to matter as the rod length approaches the Debye length.
We further note that the results for the square-tile nanostructure may be compared to those obtained for the idealized "thin disk". In order to enable such a comparison, we mapped the geometry of our square-tile on to that of a disc whose surface area corresponds to that of the square-tile and estimated an effective radius for the square-tile. In the regime where <