On the Improvement of Free-Energy Calculation from Steered Molecular Dynamics Simulations Using Adaptive Stochastic Perturbation Protocols

<div><p>The potential of mean force (PMF) calculation in single molecule manipulation experiments performed via the steered molecular dynamics (SMD) technique is a computationally very demanding task because the analyzed system has to be perturbed very slowly to be kept close to equilibrium. Faster perturbations, far from equilibrium, increase dissipation and move the average work away from the underlying free energy profile, and thus introduce a bias into the PMF estimate. The Jarzynski equality offers a way to overcome the bias problem by being able to produce an exact estimate of the free energy difference, regardless of the perturbation regime. However, with a limited number of samples and high dissipation the Jarzynski equality also introduces a bias. In our previous work, based on the Brownian motion formalism, we introduced three stochastic perturbation protocols aimed at improving the PMF calculation with the Jarzynski equality in single molecule manipulation experiments and analogous computer simulations. This paper describes the PMF reconstruction results based on full-atom molecular dynamics simulations, obtained with those three protocols. We also want to show that the protocols are applicable with the second-order cumulant expansion formula. Our protocols offer a very noticeable improvement over the simple constant velocity pulling. They are able to produce an acceptable estimate of PMF with a significantly reduced bias, even with very fast perturbation regimes. Therefore, the protocols can be adopted as practical and efficient tools for the analysis of mechanical properties of biological molecules.</p></div

PMF reconstruction quality expressed as the relative RMSD, for the reconstructions calculated with the Jarzynski equality.

<p>The pulling trajectories were generated using the three stochastic perturbation protocols (constant variance noise, AM noise and FM noise). The noise amplitude <i>m</i> was in the range between 160 and 210, for the Gaussian distribution of the pulling point (panels <i>a</i>, <i>b</i> and <i>c</i>), and between 80 and 130, for the chi-square distribution of the pulling point (panels <i>d</i>, <i>e</i> and <i>f</i>). The full line represents the reconstruction quality for the normal pulling and 10 m/s velocity. The dashed line is the reconstruction quality for the normal pulling and 1 m/s velocity.</p

Additional file 4: Table S4. of A novel joint analysis framework improves identification of differentially expressed genes in cross disease transcriptomic analysis

Overlapped genes comparison between single data set and joint analysis. (XLSX 8 kb

Additional file 1: Table S1. of A novel joint analysis framework improves identification of differentially expressed genes in cross disease transcriptomic analysis

Comparison of average sensitivity and FDR between single and joint analysis. (XLSX 16 kb

Frequency-modulated (FM) noise based reconstructions.

<p>a) Jarzynski PMF estimates based on the FM stochastic perturbation protocol. The external Gaussian noise was applied with the 10 m/s pulling velocity, using four noise amplitudes, m = 180, 190, 200 and 210. b) Second order cumulant expansion PMF estimates based on the FM stochastic perturbation protocol. The chi-square noise was applied to 10 m/s pulling velocity using 4 different noise amplitudes, <i>m</i> = 100, 110, 120 and 130. The estimates based on the normal pulling (1 m/s and 10 m/s) are given for the comparison. In each case depicted the computational cost is the same, i.e., it is analogous to the cost required to generate 10,000 trajectories using the normal pulling and 10 m/s velocity.</p

Constant variance noise based reconstructions.

<p>a) Jarzynski PMF estimates based on the constant variance noise protocol. The Gaussian noise was applied with the 10 m/s pulling velocity. Four noise amplitudes are depicted, <i>m</i> = 180, 190, 200 and 210. b) Cumulant expansion PMF estimates based on the constant variance noise protocol. The Chi-square noise was applied with the 10 m/s pulling velocity. Four noise amplitudes are depicted, <i>m</i> = 100, 110, 120 and 130. The estimates based on the normal pulling (1 m/s and 10 m/s) are given for the comparison. In each case depicted the computational cost is the same, i.e., it is analogous to the cost required to generate 10,000 trajectories using normal pulling and 10 m/s velocity.</p

Jarzynski PMF estimate fluctuations analysis.

<p>a) Extraction of the estimate fluctuations by subtracting the filtered estimate from the original Jarzynski estimate. b) Creation of the function <i>V_noise</i>(<i>r</i>) by filtering out the estimate fluctuations (as an average extracted from 5 reconstructions based on 10 different trajectories each).</p

Amplitude-modulated (AM) noise based reconstructions.

<p>a) Jarzynski PMF estimates based on the AM protocol. The Gaussian noise was applied with the 10 m/s pulling velocity, using four noise amplitudes, <i>m</i> = 180, 190, 200 and 210. b) Second order cumulant expansion PMF estimates based on the AM protocol. The chi-square noise was applied with the 10 m/s pulling velocity using 4 different noise amplitudes, <i>m</i> = 100, 110, 120 and 130. The estimates based on the normal pulling (1 m/s and 10 m/s) are given for the comparison. In each case depicted the computational cost is the same, i.e., it is analogous to the cost required to generate 10,000 trajectories using the normal pulling and 10 m/s velocity.</p

Normal pulling based estimates a) Jarzynski based PMF estimate based on the normal 10 m/s pulling and limited number of trajectories (100).

<p>The thin and “noisy” line is the estimate, and the thick line passing through it is its smoothed version. The line on the bottom is the absolute difference between the estimate and its smoothed version; it represents the estimate’s fluctuations. We used these fluctuations to analyze the behavior and determine the maximum bias of our estimates. The dashed line is PMF. The dotted line is the mean work. b) PMF estimates based on the normal pulling protocol and the Jarzynski equality, for 1 m/s and 10 m/s pulling velocities. c) PMF estimates based on the normal pulling protocol and the cumulant expansion formula, for 1 m/s and 10 m/s pulling velocities. For each pulling velocity two estimates are given, one based on the maximum number of work trajectories, and the other on 10 times less trajectories. With the slower pulling, we used fewer trajectories in order to make the comparison on equal terms, i.e., using the same computational cost.</p

PMF reconstruction quality expressed as the relative RMSD, for the reconstructions calculated with the cumulant expansion formula.

<p>The pulling trajectories were generated using the three stochastic perturbation protocols (constant variance noise, AM noise and FM noise). The noise amplitude <i>m</i> was in the range between 80 and 130, for the chi-square distribution of the pulling point (panels <i>a</i>, <i>b</i> and <i>c</i>), and between 160 and 210, for the Gaussian distribution of the pulling point (panels <i>d</i>, <i>e</i> and <i>f</i>). The full line represents the reconstruction quality for the normal pulling and 10 m/s velocity. The dashed line is the reconstruction quality for the normal pulling and 1 m/s velocity.</p