Analysis of the Results of Metadynamics Simulations by metadynminer and metadynminer3d
CC ONTRIBUTED RESEARCH ARTICLE Analysis of the Results of MetadynamicsSimulations by metadynminer andmetadynminer3d by Dalibor Trapl and Vojtˇech Spiwok
Abstract
The molecular simulations solve the equation of motion of molecular systems, making3D shapes of molecules four-dimensional by adding the time coordinate. These methods havea great potential in drug discovery because they can realistically model the structures of proteinmolecules targeted by drugs as well as the process of binding of a potential drug to its molecular target.However, routine application of biomolecular simulations is hampered by the very high computationalcosts of this method. Several methods have been developed to address this problem. One of them,metadynamics, disfavors states of the simulated system that have been already visited and thusforces the system to explore new and new states. Here we present the package metadynminer and metadynminer3d to analyze and visualize results from metadynamics, in particular those producedby a popular metadynamics package Plumed.
Introduction
Molecular simulations and their pioneers Martin Karplus, Michael Levitt, and Arieh Warshel havebeen awarded Nobel Prize in 2013 (Karplus, 2013). These methods, in particular the method ofmolecular dynamics simulation, computationally simulate the motions of atoms in a molecularsystem. A simulation starts from a molecular system defined by positions (Cartesian coordinates) ofindividual atoms. The heart of the method is in calculation of forces acting on individual atoms andtheir numerical integration in the spirit of Newtonian dynamics, i.e., conversion of a force vector toacceleration vector, velocity vector and, finally, to a new position of an atom. By repeating these steps,it is possible to reconstruct a record of atomic motions known as a trajectory.Molecular simulations have a great potential in drug discovery. A molecule of drug influences(enhances or blocks) the function of some biomolecule in the patient’s body, typically a receptor,enzyme or other protein. These molecules are called drug targets. The process of design of a new drugcan be significantly accelerated by knowledge of the 3D structure (Cartesian coordinates of atoms) ofthe target. With such knowledge, it is possible to find a "druggable" cavity in the target and a moleculethat fits and favourably binds to this cavity to influence its function. Strong binding implies that thedrug influences the target even in low doses, hence does not cause side effects by interacting withunwanted targets.Experimental determination of 3D structures of proteins and other biomolecules is very expensiveand laborious process. Molecular simulations can, at least in principle, replace such expensive andlaborious experiments by computing. In principle, a molecular simulation starting from virtually any3D shape of a molecule would end up in energetically the most favourable shape. This is analogouswith water flowing from mountains to valleys and not in the opposite way.Unfortunately, this approach is extremely computationally expensive. The integration step of asimulation must be small enough to comprise the fastest motions in the molecular system. In practicalsimulations, it is necessary to use femtosecond integration steps. This means that it is necessary tocarry out thousands of steps to simulate picoseconds, millions of steps to simulate nanoseconds, andso forth. In each step, it is necessary to evaluate a substantial number of interactions between atoms.As the result of this, it is possible to routinely simulate nano- to microseconds. Longer simulationsrequire special high-performance computing resources.Protein folding, i.e., the transition from a quasi-random to the biologically relevant 3D structure,takes place in microseconds for very small proteins and in much longer time scales for pharmaceuticallyinteresting proteins. For this reason, prediction of a 3D structure by molecular simulations is limitedto few small and fast folding proteins. For large proteins it is currently impossible or at least far frombeing routine.Several methods have been developed to address this problem. Metadynamics (Laio and Parrinello,2002) uses artificial forces to force the system to explore states that have not been previously exploredin the simulation. At the beginning of the simulation, it is necessary to chose some parameters of thesystem referred to as collective variables. For example, numerically expressed compactness of theprotein can be used as a collective variable to accelerate its folding from a noncompact to a compact3D structure. Metadynamics starts as a usual simulation. After a certain number of steps (typically
The R Journal Vol. XX/YY, AAAA 20ZZ ISSN 2073-4859 a r X i v : . [ q - b i o . B M ] A ug ONTRIBUTED RESEARCH ARTICLE metadynminer and metadynminer3d . Metadynminer and metadynminer3d use the results of metadynamics simulations to calculatethe free energy surface of the molecular system. The most favoured states (states most populated inreality) correspond to minima on the free energy surface. The state with the lowest free energy is themost populated state in the reality, i.e., the folded 3D structure of the protein.As an example to illustrate metadynamics and our package, we use an ultrasimple molecule of"alanine dipeptide" (Figure 1). This molecule can be viewed as a "protein" with just one amino acidresidue (real proteins have hundreds or thousands of amino acid residues). As a collective variableit is possible to use an angle φ defined by four atoms. Biasing of this collective variable acceleratesa slow rotation around the corresponding bond. Figure 1 shows the free energy surface of alaninedipeptide as the black thick line. It is not known before the simulation. The simulation starts from thestate B. After 500 simulation steps, the hill is added (the hill is depicted as the red line, the floodingpotential ("sand") at the top, the free energy surface with added flooding potential at the bottom). Sumof 10, 100, 200, 500, and 700 hills is depicted as yellow to blue lines.At the end of simulation the free energy surface is relatively well flattened (blue line in Fig. 1bottom). Therefore, the free energy surface can be estimated as a negative imprint of added "sand": G ( s ) = − kT log ( P ( s )) = − V ( s ) = ∑ i w i exp ( − ( s − S i ) /2 σ ) (1)where G , V , and P are free energy, metadynamics bias (flooding) potential, and probability, respectively,of a state with a collective variable s , k is Boltzmann constant, T is temperature in Kelvins, w i is height, S i is position and σ i is width of each hill. The equation can be easily generalized for two or more CVs.The original version of metadynamics was developed with constant heights of Gaussian hills. Later,a so-called well-tempered metadynamics was developed (Barducci et al., 2007), which uses decreasinghill heights to improve the accuracy of the results. This requires modification of the equation: G ( s ) = − kT log ( P ( s )) = − T + ∆ T ∆ T V ( s ) = − T + ∆ T ∆ T ∑ i w i exp ( − ( s − S i ) /2 σ ) (2)where ∆ T an input parameter with the dimension of temperature (zero for unbiased simulation andinfinity for the original metadynamics with constant hill heights). Nowadays, the vast majority ofmetadynamics applications use the well-tempered metadynamics algorithm for its better convergencetowards accurate free energy surface prediction.There are numerous packages for molecular simulations such as Amber (Weiner and Kollman,1981), Gromacs (Abraham et al., 2015), Gromos (Christen et al., 2005), NAMD (Phillips et al., 2020),CHARMM (Brooks et al., 2009), Acemd (Harvey et al., 2009), and others. These packages are primarilydeveloped for basic unbiased simulations with no or very limited support of metadynamics. Plumedsoftware (Tribello et al., 2014) has been developed to introduce metadynamics into various simulationprograms. Since its introduction, Plumed articles have been cited in more than thousand papers fromdrug design, molecular biology, material sciences, and other fields. The R package metadynminer was developed for analysis and visualization of the results from Plumed. With a simple file conversionscript, it can be used also with other simulation programs that support metadynamics. Example of usage
Metadynminer will be presented on a bias potential from a 30 ns (30,000 hills) simulation of alaninedipeptide (Figure 1). Two rotatable bonds of the molecule, referred to as φ and ψ , were used ascollective variables. This is basically an expansion of the free energy surface in Figure 1 to two dimen- The R Journal Vol. XX/YY, AAAA 20ZZ ISSN 2073-4859
ONTRIBUTED RESEARCH ARTICLE Figure 1:
Metadynamics simulation of alanine dipeptide. Dihedral angle φ was used as the collectivevariable. The top part shows molecular structures of three free energy minima (stable structures)differing in the value of φ . According to metadynamics prediction, A is the global minimum (freeenergy 0 kJ/mol) and B and C are local minima (1.5 and 6.3 kJ/mol, respectively). According toEquation 1, this corresponds to probabilities 0.61, 0.34, and 0.05 for A, B, and C, respectively. Themiddle part shows the bias potential (scaled by ( T + ∆ T ) / ∆ T ) after addition of 1, 10, 100, 200, 500, and700 hills (colors from red to blue). The bottom part shows the accurate free energy surface calculatedby metadynamics with 30,000 hills (black) flooded by 1, 10, 100, 200, 500, and 700 hills (colors fromred to blue). The figure was generated by metadynminer except for molecular structures and finalassembly. The R Journal Vol. XX/YY, AAAA 20ZZ ISSN 2073-4859
ONTRIBUTED RESEARCH ARTICLE sions. Hills from simulations with two collective variables ( φ and ψ ) and with one collective variable( φ ) are provided in metadynminer as acealanme and acealanme1d , respectively. Metadynminer3d was developed for analysis of metadynamics with three collective variables. It contains a sampledata acealanmed3 , with collective variables φ , ψ and ω . We decided to distribute metadynminer and metadynminer3d separately, because of the use of different visualization tools and to keep thesize of packages low. Metadynamics simulations with 1-3 CVs comprise almost all metadynamicsapplications nowadays (not considering special metadynamics variants).Hills file generated by Plumed package (filename "HILLS") can be loaded to R by the function read.hills : hillsfile <- read.hills("HILLS", per=c(T, T)) The parameter per indicates periodicity of the collective variable (dihedral angles are periodic, i.e., + π (cid:39) − π ).Typing the name hillsfile will return its dimensionality (the number of CVs) and the number ofhills. A hills object can be easily plotted: plot(hillsfile, xlab="phi", ylab="psi") For metadynamics with one collective variable, it plots its evolution. For metadynamics with two orthree collective variables, it plots a scatter plot of collective variables number 1 vs. 2 or 1 vs. 2 vs. 3,respectively (Figure 2).
Figure 2:
Scatter plot of hills position.In well-tempered metadynamics it may be interesting to see the evolution of hill heights ( w i inEquation 2). This can be plotted (Figure 3) by typing: plotheights(hillsfile) Addition operation is available for hillsfile object. For example, multiple hills files can be concate-nated.Next, the user can sum negative values of all hills to make the free energy surface estimate bytyping: fesurface <- fes(hillsfile)
Hills files from well-tempered metadynamics are prescaled by ( ∆ T + T ) / ∆ T when printed by Plumed,so no special action is required in metadynminer . The function fes uses the Bias Sum algorithm(Hošek and Spiwok, 2016). This function is fast because instead of evaluation of Gaussian function forevery hill, it uses a precomputed Gaussian hill that is relocated to hill centers. It is also fast because itwas implemented in C++ via Rcpp . Because of approximations used in the function fes , this functionshould be used for visualization purposes. For detailed analysis of a free energy surface, we advice touse a slow but accurate fes2 function. This function explicitly evaluates Gaussian function for everyhill. It can be also used for (rarely used) metadynamics with variable hill widths.Typing the name of the variable with a free energy surface returns its dimensionality, numberof points, and free energy maximum and minimum. The same is returned by summary function. It
The R Journal Vol. XX/YY, AAAA 20ZZ ISSN 2073-4859
ONTRIBUTED RESEARCH ARTICLE Figure 3:
Evolution of heights of hills in metadynamics plotted by function plotheights .is possible to add and subtract two free energy surfaces with the same number of grid points. Thefunctions min and max can be used as well to calculate minimum or maximum. It is also possible tomultiply or divide the free energy surface by a constant (for example, to convert kJ to kcal and viceversa). Free energy surface can be plotted (Figure 4) by typing: plot(fesurface, xlab="phi", ylab="psi")
Figure 4:
Free energy surface.In metadynamics simulation, it is important to find free energy minima. The global minimumrefers to the most favored state of the system (i.e., the state with the highest probability). Other localminima correspond to metastable states. The user can find free energy minima by typing: minima <- fesminima(fesurface)
This function locates minima using a simple algorithm. The free energy surface is separated into 8, 8x8,or 8x8x8 bins (for 1D, 2D, or 3D surface, respectively). A minimum in each bin is located. Next, theprogram tests whether a minimum is a local minimum of the whole free energy surface. The numberof grid points can be changed by ngrid parameter. Typing the name of the minima variable will returnthe table of minima (denoted as A, B, C, ... in the order of their free energies), their collective variables,and free energy values.In addition, the function summary provides populations of each minimum calculated as: P i , rel = exp ( − G i / kT ) (3) The R Journal Vol. XX/YY, AAAA 20ZZ ISSN 2073-4859
ONTRIBUTED RESEARCH ARTICLE P i = P i , rel / ∑ ( P j , rel ) (4) letter CV1bin CV2bin CV1 CV2 free_energy relative_pop1 A 78 236 -1.2443171 2.6487938 -97.26095 8.614856e+162 B 28 240 -2.4763142 2.7473536 -95.63038 4.480527e+163 C 74 118 -1.3428769 -0.2587194 -94.73163 3.124915e+164 D 166 151 0.9239978 0.5543987 -91.66626 9.143024e+155 E 170 251 1.0225576 3.0183929 -84.37799 4.920882e+14pop1 50.13356582 26.07412013 18.18522684 5.32072005 0.2863674 Plot function on a fesminima output provides the same plot as for fes output with additionalletters indicating minima (Figure 5).
Figure 5:
Free energy surface with indicated free energy minima A-E.It is essential to evaluate the accuracy of metadynamics and to decide when the simulation isaccurate enough so that it can be stopped. For this purpose, it is useful to look at the evolution ofrelative free energies. The relative free energies (for example, the free energy difference of minimaA and C) evolve rapidly at the beginning of the simulation, and with the progress of the simulation,their difference is converging towards the real free energy difference. Function feprof calculates theevolution of free energy differences from the global minimum (global at the end of the simulation). Itcan be used as: prof<-feprof(minima)
Function summary provides minima and maxima of these free energy differences. The evolution canbe plotted (Figure 6) by typing: plot(prof)
Beside minima, another important points on the free energy surface are transition states. Changeof the molecular structure from one minimum to another takes place via a path with the lowest energydemand. The state with the highest energy along this path is called the transition state. Free energydifference between the initial and transition state can be used to predict kinetics (rates) of the studiedmolecular process. Furthermore, identification of transition states is important in drug design becausecompounds designed to mimic a transition states of an enzymatic reaction are often potent enzymesinhibitors and drugs (Itzstein et al., 1993).In metadynminer , such path can be identified by Nudged Elastic Band method (Henkelman andJónsson, 2000). Briefly, this method plots a line between selected minima as an initial approximationof the transition path. Next, this line is curved so that the corresponding physical process becomesfeasible. This function can be applied on, for example, minima A and D as:
The R Journal Vol. XX/YY, AAAA 20ZZ ISSN 2073-4859
ONTRIBUTED RESEARCH ARTICLE Figure 6:
Evolution of free energy differences. nebAD<-neb(minima, min1="A", min2="D")
The result can be analyzed by summary (to provide kinetics of the A to D and D to A change predictedby Eyring equation (Eyring, 1935)), by plot (to plot the free energy profile of the molecular process)and by pointsonfes of linesonfes (to plot the path on top of the free energy surface). The last examplecan be invoked by: plot(minima, xlab="phi", ylab="psi")linesonfes(nebAD, lwd=4) The resulting plot is depicted in Figure 7
Figure 7:
Path of transition from A to D projected onto free energy surface.Let us also briefly present metadynminer3d . This package uses packages rgl and misc3d to plotthe free energy surface as an interactive (mouse rotatable) isosurface in the space of three collectivevariables (see Figure 8).
Metadynminer3d can produce interactive WebGL visualizations using writeWebGL command from the rgl package.
Metadynminer and metadynminer3d were developed to be highly flexible. This flexibility canbe demonstrated on two examples. First, it is useful to visualize the progress of metadynamics as avideo sequence showing the evolution of the free energy surface. The code to generate correspondingimages can be written in metadynminer as: tfes<-fes(acealanme, tmax=100)png("snap%04d.png")
The R Journal Vol. XX/YY, AAAA 20ZZ ISSN 2073-4859
ONTRIBUTED RESEARCH ARTICLE Figure 8:
3D free energy surface depicted as isosurface at −
30 kJ/mol. plot(acealanme, zlim=c(-200,0))for(i in 1:299) {tfes<-tfes+fes(acealanme, imin=100*i+1, imax=100*(i+1))plot(tfes, zlim=c(-200,0), xlab="phi", ylab="psi")}dev.off()
This generates a series of images that can be concatenated by external software to make a video file.The second example demonstrates a more complicated analysis of the results from metadynamics.Functions fes and fes2 use equations 1 and 2 to predict the free energy surface. A limitation of thisapproach is that the prediction of the free energy surface is based only on the positions of hills. Theevolution of collective variables between hills depositions is not used. As an alternative, it is possibleto use reweighting (Torrie and Valleau, 1977; Tiwary and Parrinello, 2015). This approach calculatesthe free energy surface from hills positions as well as from evolution of collective variables. Briefly,regions of the free energy surface that are sampled despite being disfavored by high flooding potentialhave higher weights than those disfavored by low flooding potential. This approach is in general moreaccurate. Reweighting can be done using the code: bf <- 15kT <- 8.314*300/1000npoints <- 50maxfes <- 75outfes <- 0*fes(acealanme, npoints=npoints)s1 <- c()s2 <- c()for(i in 1:50) {step <- i*length(acealanme$time)/50cfes <- fes(acealanme, imax=step)s1 <- c(s1, sum(exp(-cfes$fes/kT)))s2 <- c(s2, sum(exp(-cfes$fes/kT/bf)))}ebetac <- s1/s2cvs <- read.table("COLVAR")nsamples <- nrow(cvs)xlim <- c(-pi,pi)ylim <- c(-pi,pi)for(i in 1:nsamples) {step <- (i-1)*50/nsamples+1ix <- npoints*(cvs[i,2]-xlim[1])/(xlim[2]-xlim[1])+1iy <- npoints*(cvs[i,3]-ylim[1])/(ylim[2]-ylim[1])+1outfes$fes[ix,iy] <- outfes$fes[ix,iy] + exp(cvs[i,4]/kT)/ebetac[step]}outfes$fes[!outfes$fes>0] <- maxfesoutfes$fes <- -kT*log(outfes$fes)
The R Journal Vol. XX/YY, AAAA 20ZZ ISSN 2073-4859
ONTRIBUTED RESEARCH ARTICLE plot(outfes, xlab="phi", ylab="psi") where bf is the bias factor ( ( T + ∆ T ) / T in Equation 2), kT is temperature in Kelvins multiplied byBoltzmann constant, npoints is the granularity of the resulting free energy surface and maxfes is themaximal possible free energy (to avoid problems with infinite free energy in unsampled regions).First, outfes is introduced as a zero free energy surface. The first loop calculates the correction ebetac for the evolution of flooding potential developed by Tiwary and Parrinello (Tiwary and Parrinello,2015). Next, a file with the evolution of collective variables COLVAR (from the same simulation usedto generate acealanme dataset, available at ) isread. The second loop evaluates the sampling weighted by the factor exp ( V ( s ) / kT ) divided by ebetac to correct for the evolution of the bias potential (Tiwary and Parrinello, 2015). Finally, probabilities areconverted to the free energy surface and plotted (Figure 9). Figure 9:
Free energy surface calculated by reweighting by Tiwary and Parrinello (2015).
Simulation details
All simulations were done using Gromacs 2016.4 (Abraham et al., 2015) patched by Plumed 2.4b(Tribello et al., 2014). Alanine dipeptide was modeled using Amber99SB-ILDN force field (Lindorff-Larsen et al., 2010). The simulated system contained alanine dipeptide and 874 TIP3P (Jorgensen et al.,1983) water molecules. The temperature was kept constant at 300 K using Bussi thermostat (Bussi et al.,2007). Metadynamics hills of height 1 kJ/mol (bias factor 10) and widths 0.3 rad were added every 1 ps.Two simulations were done, one with one dihedral angle φ (dataset acealanme1d ), two dihedral angles φ and ψ (dataset acealanme ), or with three angle φ , φ and ω (dataset acealanme3d in metadynminer3d ).Supporting material is available at or in Plumednest (PLUMED consortium, 2019) at . Summary
The package metadynminer and metadynminer3d provides fast algorithm Bias Sum (Hošek andSpiwok, 2016) for calculation of free energy surfaces from metadynamics. This algorithm is availablein our on-line tool MetadynView ( http://metadyn.vscht.cz ), but this tools is intended for routinechecking of the progress of metadynamics simulations rather than for in deep analysis and visualiza-tion. Besides this, users of metadynamics use built-in functions in Plumed or various in-lab scripts.Such scripts do not provide appropriate flexibility in analysis and visualization.We see the biggest advantage in the fact that metadynminer can produce publication qualityfigures via graphics output functions in R. As shown above, using a simple for loop it is possibleto plot individual snapshots and concatenate them outside R to make a movie.
Metadynminer3d provides the possibility to produce interactive 3D web models by WebGL technology. We also tested3D printing of a free energy surface that is very easy using metadynminer and rayshader . Various tipsand tricks can be found on the website of the project ( ). The R Journal Vol. XX/YY, AAAA 20ZZ ISSN 2073-4859
ONTRIBUTED RESEARCH ARTICLE Another advantage we see in reporting of results. Reproducibility is a big issue in science,including molecular simulations. Packages like knitr or rmarkdown ca be used to record all steps ofdata analysis pipeline to compile a report for routine and reproducible use of metadynamics. Acknowledgement
This project was supported by Ministry of Education, Youth and Sports of the Czech Republic - COSTaction OpenMultiMed (CA15120, LTC18074) for development and Czech National Infrastructure forBiological data (ELIXIR CZ, LM2015047) for future sustainability.
Bibliography
J. M. Abraham, T. Murtola, R. Schulz, S. Páll, J. C. Smith, B. Hess, and L. Erik. Gromacs: Highperformance molecular simulations through multi-level parallelism from laptops to supercomputers.
SoftwareX , 1–2:19–25, 2015. URL https://doi.org/10.1016/j.softx.2015.06.001 . [p2, 9]A. Barducci, G. Bussi, and M. Parrinello. Well-tempered metadynamics: A smoothly converging andtunable free-energy method.
Physical Review Letters , 100(2):020603, 2007. URL https://doi.org/10.1103/PhysRevLett.100.020603 . [p2]B. R. Brooks, C. L. Brooks, A. D. Mackerell, L. Nilsson, R. J. Petrella, B. Roux, Y. Won, G. Archontis,C. Bartels, S. Boresch, A. Caflisch, L. Caves, Q. Cui, A. R. Dinner, M. Feig, S. Fischer, J. Gao,M. Hodoscek, W. Im, K. Kuczera, T. Lazaridis, J. Ma, V. Ovchinnikov, E. Paci, R. W. Pastor, C. B. Post,J. Z. Pu, M. Schaefer, B. Tidor, R. M. Venable, H. L. Woodcock, X. Wu, W. Yang, D. M. York, andM. Karplus. Charmm: The biomolecular simulation program.
Journal of Computational Chemistry , 30(10):1545–1614, 2009. URL https://doi.org/10.1002/jcc.21287 . [p2]G. Bussi, D. Donadio, and M. Parrinello. Canonical sampling through velocity rescaling.
Journal ofChemical Physics , 1(126):014101, 2007. URL https://doi.org/10.1063/1.2408420 . [p9]M. Christen, P. H. Hünenberger, D. Bakowies, R. Baron, R. Bürgi, D. P. Geerke, T. N. Heinz, M. A.Kastenholz, V. Kräutler, C. Oostenbrink, C. Peter, D. Trzesniak, and W. F. van Gunsteren. Thegromos software for biomolecular simulation: Gromos05.
Journal of Computational Chemistry , 26(16):1719–1751, 2005. URL https://doi.org/10.1002/jcc.20303 . [p2]H. Eyring. The activated complex in chemical reactions.
The Journal of Chemical Physics , 3(2), 1935.URL https://doi.org/10.1063/1.1749604 . [p7]M. J. Harvey, G. Giupponi, and G. D. Fabritiis. Acemd: Accelerating biomolecular dynamics in themicrosecond time scale.
Journal of Chemical Theory and Computation , 5(6):1632–1639, 2009. URL https://doi.org/10.1021/ct9000685 . [p2]G. Henkelman and H. Jónsson. Improved tangent estimate in the nudged elastic band method forfinding minimum energy paths and saddle points.
The Journal of Chemical Physics , 113(22):9978–9985,2000. URL https://doi.org/10.1063/1.1323224 . [p6]P. Hošek and V. Spiwok. Metadyn view: Fast web-based viewer of free energy surfaces calculated,by metadynamics.
Computer Physics Communications , 198:222–229, 2016. URL https://doi.org/10.1016/j.cpc.2015.08.037 . [p4, 9]M. v. Itzstein, W.-Y. Wu, G. B. Kok, M. S. Pegg, J. C. Dyasson, B. Jin, T. V. Phan, M. L. Smythe, H. F.White, S. W. Oliver, P. M. Colman, J. N. Varghese, D. M. Ryan, J. M. Woods, R. C. Bathell, V. J.Hotham, J. M. Cameron, and C. R. Penn. Rational design of potent sialidase-based inhibitors ofinfluenza virus replication.
Nature , 363:418–423, 1993. URL https://doi.org/10.1038/363418a0 .[p6]W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey, and M. L. Klein. Comparison of simplepotential functions for simulating liquid water.
The Journal of Chemical Physics , 79(2):926, 1983. URL https://doi.org/10.1063/1.445869 . [p9]M. Karplus. Nobel lecture, 2013. URL . [p1]A. Laio and M. Parrinello. Escaping free-energy minima.
Proceedings of the National Academy of Sciencesof the United States of America , 20(99):12562–12566, 2002. URL https://doi.org/10.1073/pnas.202427399 . [p1]
The R Journal Vol. XX/YY, AAAA 20ZZ ISSN 2073-4859
ONTRIBUTED RESEARCH ARTICLE K. Lindorff-Larsen, S. Piana, K. Palmo, P. Maragakis, J. L. Klepeis, R. O. Dror, and D. E. Shaw. Improvedside-chain torsion potentials for the amber ff99sb protein force field.
Proteins: Structure, Function,and Bioinformatics , 78(8):1950–1958, 2010. URL https://doi.org/10.1002/prot.22711 . [p9]J. C. Phillips, D. J. Hardy, J. D. C. Maia, J. E. Stone, J. V. Ribeiro, R. C. Bernardi, R. Buch, G. Fiorin,J. Hénin, W. Jiang, R. McGreevy, M. C. R. Melo, B. K. Radak, R. D. Skeel, A. Singharoy, Y. Wang,B. Roux, A. Aksimentiev, Z. Luthey-Schulten, L. V. Kalé, K. Schulten, C. Chipot, and E. Tajkhorshid.Scalable molecular dynamics on cpu and gpu architectures with namd.
Journal of Chemical Physics ,153(4):044130, 2020. URL https://doi.org/10.1063/5.0014475 . [p2]PLUMED consortium. Promoting transparency and reproducibility in enhanced molecular simulations.
Nature Methods , 16:670–673, 2019. URL https://doi.org/10.1038/s41592-019-0506-8 . [p9]P. Tiwary and M. Parrinello. A time-independent free energy estimator for metadynamics.
The Journalof Physical Chemistry B , 119(3):736–742, 2015. URL https://doi.org/10.1021/jp504920s . [p8, 9]G. M. Torrie and J. P. Valleau. Nonphysical sampling distributions in monte carlo free-energy es-timation: Umbrella sampling.
Journal of Computational Physics , 23:187–199, 1977. URL https://doi.org/10.1016/0021-9991(77)90121-8 . [p8]G. A. Tribello, M. Bonomi, D. Branduardi, C. Camilloni, and G. Bussi. Plumed 2: New feathers for anold bird.
Computer Physics Communications , 185(2):604–613, 2014. URL https://doi.org/10.1016/j.cpc.2013.09.018 . [p2, 9]P. K. Weiner and P. A. Kollman. Amber: Assisted model building with energy refinement. a generalprogram for modeling molecules and their interactions.
Journal of Computational Chemistry , 2(3):287–303, 1981. URL https://doi.org/10.1002/jcc.540020311 . [p2]
Dalibor TraplDepartment of Biochemistry and Microbiology, University of Chemistry and Technology, PragueTechnicka 3, Prague 6, 166 28Czech RepublicORCID: 0000-0002-3435-5841 [email protected]
Vojtech SpiwokDepartment of Biochemistry and Microbiology, University of Chemistry and Technology, PragueTechnicka 3, Prague 6, 166 28Czech RepublicORCID: 0000-0001-8108-2033 [email protected]@vscht.cz