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## Results: 7

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2.

Diffusion coefficients

*D*_{app}(*L*) of A_{3}RNA as a function of the inverse box length, 1/*L*(*circles*). The solid line was obtained by fitting Eq. 3 to the calculated diffusion coefficients. The inset shows the corresponding plot for a single K^{+}ion in water. Error bars (mean ±1 standard deviation) were estimated from block averages.3.

Position

*z*(*t*) of the geometric center of A_{6}as a function of time along the direction of the electric field (*E*= 50 mV/Å). The inset is a magnified view of*z*(*t*) during a 300-ps time interval. The three straight lines in the inset are the best linear fits for three consecutive 100-ps blocks, with slopes of 18.37, 29.15, and 19.96 m s^{−1}, respectively. The average number of ions bound to the RNA during those 100-ps time intervals are 0.50, 0.03, and 0.14, respectively.4.

Variance of displacements

*z*(*t*) along the direction of the electric field (*E*= 50 mV/Å) for A_{6}RNA as a function of time*t*(*solid line*). Circles represent the results of fitting Eq. 11. Also shown are the variance of the displacement*z*(*t*) of A_{6}without electric field (*dotted line*), and variances of A_{6}in directions*x*and*y*normal to the electric field with dashed and dot-dashed lines, respectively. Differences between the variances in*x*(*t*) and*y*(*t*) at finite electric field and*z*(*t*) at zero field are within the statistical uncertainties. The inset shows a magnified view of the initial nonlinear diffusive spread.5.

(

*a*) Average drift velocity of A_{6}plotted as a function of the average number of bound counterions*N*_{K+}, calculated for 100-ps time intervals. Circles, squares, and diamonds represent the results at*E*= 50, 40, and 30 mV/Å, respectively. Solid, dotted, and dashed lines represent the drift velocities predicted by the modified Nernst-Einstein formula, Eq. 13, for*E*= 50, 40, and 30 mV/Å, respectively, calculated with the diffusion coefficient of A_{6}RNA in the MD simulation without counterions,*D*= 2.86 × 10^{−6}cm^{2}s^{−1}. (*b*) Probability distributions of the average number of bound counterions during 100-ps time intervals.6.

Probability distributions of displacements

*z*(*t*) of K^{+}ions during time intervals of*t*= 100 ps in the A_{6}/K^{+}/water solution at 0, 3, 30, 40, and 50 mV/Å shown as lines with circles, squares, diamonds, solid triangles, and crosses, respectively. Vertical arrows indicate the positions expected from free ionic drift at fields of 30, 40, and 50 mV/Å with a mobility of*D*_{K}^{+}*e*/*k*_{B}*T*, where*D*_{K}^{+}= 3.515 × 10^{−5}cm^{2}s^{−1}is the diffusion coefficient of a free K^{+}ion estimated from the inset in Fig. 1 for the system size of the A_{6}simulation. The horizontal arrow indicates the expected average drift position of a K^{+}ion bound to A_{6}RNA, −4*e D*_{RNA}*Et*/*k*_{B}*T*, at*E*= 40 mV/Å, where*D*_{RNA}is the diffusion coefficient*D*_{app}(*L*) of A_{6}listed in Table 3.7.

(

*a*) Cosine of the angle between water dipoles and the field direction as a function of the electric field*E*. Open circles connected with a dotted line represent results obtained from simulations of bulk TIP3P water. Open squares at 0, 3, 30, 40, and 50 mV/Å show the results calculated from water molecules at least 20 Å away from the RNA center in A_{6}/water simulations at corresponding electric field strengths*E*_{0}. (*b*) Polarization (*P*) of bulk TIP3P water as a function of the electric field*E*(*open circles*connected with*dotted line*). The solid line is the linear-response result (Neumann, 1983),*P*= (*ɛ*− 1)*ɛ*_{0}*E*, for a dielectric constant of*ɛ*= 94 that was obtained from a linear fit at small electric fields (*inset*), where*ɛ*_{0}is the vacuum permittivity. The nonlinearity between*P*and*E*becomes pronounced at electric fields above*E*∼ 7 mV/Å, resulting in an apparent decrease of*ɛ*(Yeh and Berkowitz, 1999a).