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Free Energy Methods in MD-Cournia-1.7 Free Energy Methods applied in Biomolecular Simulations & Drug Design Zoe Cournia Biomedical Research Founda2on, Academy of Athens 1 July 2020 [email protected] twier: @zoecournia 1 MD Simulator .me limitaons Need enhanced sampling simulations Biomolecular events depend on ΔG, T Sampling problem Free energy Almost all measurable proper7es depend (directly or indirectly) on the free energy of the system Hamiltonian (energy funcon) Free Energy Probability of finding the How the system system in a given state preferen.ally evolves in .me - conformaonal equilibria - rates of reac.on - associaon constants - transi.on paths (equilibrium proper-es) (non-equilibrium processes) Dr Paraskevi Gkeka - 02/06/14 Zoe Cournia – Academy of Athens Enhanced sampling techniques Out-of-equilibrium methods that bias the system to explore regions of high free energy When they are used: •Large systems (Millions of atoms) •Long simulaon .me (extensive sampling) •Exploraon of rough energy landscapes (rare events) Detachment of nSH2 from helical domain Phenomena that can be studied: •Large protein conformaonal changes •Protein – ligand binding •Protein folding •Protein – protein interac.ons •Protein – membrane associaon •Transport through membranes 6 Replica exchange • Parallel MD simulaons (replicas) carried out at different temperatures (physiological 300 K and higher) • The increased temperatures allow for the system to overcome energe.c barriers and sample more phase space • Exchange of configuraons between the replicas Detachment of nSH2 from helical domain R.C. Bernardi, et al., Enhanced sampling techniques in molecular dynamics simulaons of biological systems, Biochim. 7 Biophys. Acta (2014), h_p://dx.doi.org/10.1016/j.bbagen.2014.10.019 Steered Molecular Dynamics (SMD) Pulling of one or more CVs The inial and final values of each CV are defined Be carefull!! Huge error in the calculaon of 8 Crooks Theorem - SMD We need many trajectories to create the work distribu.ons! BUT we can use directly the difference of the forward and backward work values to get a rough es.maon of the ΔF. 9 Biased SMD to force the nSH2 detachment The free energy barrier for the nSH2 domain detachment is lower in the mutant than in the WT protein BUT: Calculaon not converged WT Mutant From 10ns runs: deltaF = 353,10 ± 7,8 KJ/mol From 10ns runs: deltaF = 271,70 ± 7,7 KJ/mol From 5ns runs: deltaF = 413,78 ± 11,3 KJ/mol From 5ns runs: deltaF = 369,27 ± 7,8 KJ/mol Umbrella Sampling Is there a way to observe the NP penetra7ng the membrane with MD simulaons? Umbrella sampling is a technique used to improve sampling where ergodicity is hindered by the form of the system's energy landscape • A harmonic poten7al W is applied to restrain NP’s posi7on at different points (windows). • Results from different windows are combined and W is removed using a sta7s7cal method called WHAM • Poten7al of Mean Force (PMF) is calculated Investigating nanoparticle - membrane penetration for enhanced drug delivery The fully hydrophobic nanoparticle 0 ns 114 ns 115 ns 116 ns 117 ns 2000 ns Dr Paraskevi Gkeka - 02/06/14 The striped and the random nanopar.cle Different posi.on adopted for the random and Paraskevi Gkeka striped nanopar.cles in this simple representaon Simulaon snapshots at t=1µs random striped Zoe Cournia – Academy of Athens Nanopar.cles successfully deliver RNA interference in cancer paents Free energy calculaons Umbrella sampling: harmonic potenal W is applied to restrain NP’s posion. § 4096 DPPC + 120,541 CG waters z1 z2 Bilayer center § Hydrophobic + hydrophilic NP § NPT (T=325K), PBC § 50 windows with 1 Å spacing § Independent ini.al configuraons § 100 ns per window (75ns equilibraon) § 3 different sets of simulaons § Force constant 1000 kJ mol -1 nm-2 § the nanopar.cles are free to rotate z §Weighted histogram analysis method Water phase (WHAM) for calculaon of PMF Dr Paraskevi Gkeka - 02/06/14 Zoe Cournia – Academy of Athens Nanopar.cles successfully deliver RNA interference in cancer paents Free energy calculaons Is there a way to observe the NP penetra7ng the membrane with MD simulaons? Umbrella sampling is a technique used to improve sampling where ergodicity is hindered by the form of the system's energy landscape • A harmonic poten7al W is applied to restrain NP’s posi7on at different points (windows). • Results from different windows are combined and W is removed using a sta7s7cal method called WHAM • Poten7al of Mean Force (PMF) is calculated Nanopar.cles successfully deliver RNA interference in cancer paents Free energy calculaons Is there a way to predict possible modes of interaction between nanoparticles and membranes? z1 z2 Bilayer center ) kT striped Free energy ( random z Agreement with the eXperimental Water phase studies of Stellaci et al. Dr Paraskevi Gkeka - 02/06/14 Zoe Cournia – Academy of Athens The “hairy” nanopar.cle Core diameter = 4.3 nm 2:1 MUS:OT rao Verma et al. Nature Materials 2008 Zoe Cournia – Academy of Athens Partition mechanism in membranes Snorkeling Six different systems were considered: effect - 0%, 10%, 20%, 30%, 40%, and 50% mol. cholesterol - 8,000-10,000 lipids - Simulation time ~10 µs – No insertion was observed - 5 µs with the NP already inside – Snorkeling effect Dr Paraskevi Gkeka - 02/06/14 Free energy calculations Is there a way to observe the NP penetra7ng the membrane with MD simulaons? Umbrella sampling is a technique used to improve sampling where ergodicity is hindered by the form of the system's energy landscape • A harmonic poten7al W is applied to restrain NP’s posi7on at different points (windows). • Results from different windows are combined and W is removed using a sta7s7cal method called WHAM • Poten7al of Mean Force (PMF) is calculated Umbrella sampling calculaons PMF for NP par..oning in a cholesterol-free DPPC lipid bilayer and a bilayer containing 50% mol. cholesterol. The error bars have been calculated using the bootstrap method. ΔΔG0% = 25.8± 2.1kJ / mol 323 K, 50% chol ΔΔG50% = 56.1±1.9kJ / mol 323 K, 0% chol Gkeka et al, PLOS Comput Biol (2014) Dr Paraskevi Gkeka - 02/06/14 Zoe Cournia – Academy of Athens Free energy calculaons 8 nm z1 z2 Bilayer center § anionic NP, 0% and 50% mol. Chol § 2 independent simulaons per PMF § four subsequent annealing simulaons at 450K, 400K, 380K, and finally 323K § NPT (T=325K), PBC § 40 windows with 2 Å spacing § 150 ns per window (100 ns equilibraon) § Force constant 7500 kJ mol -1 nm-2 z §Weighted histogram analysis method Water phase (WHAM) for calculaon of PMF DrZoe Paraskevi Cournia –Gkeka Academy - 02/06/14 of Athens Convergence of umbrella sampling calculaons Barrier is essenally the same when we use different sampling intervals Dr Paraskevi Gkeka - 02/06/14 Zoe Cournia – Academy of Athens Convergence of umbrella sampling calculaons Dr Paraskevi Gkeka - 02/06/14 Zoe Cournia – Academy of Athens Metadynamics • Energy is provided to the system in order to fill the energy wells • Defini.on of Collec.ve Variables (CVs), i.e. proper.es of the system, to reduce the dimensionality of the system • Choice of CVs: 1. Dis.nguish between the ini.al, final, and intermediate states clearly 2. describe all the slow events that are relevant to the process of interest Detachment of nSH2 from helical domain 3. Their number should not be too large, otherwise it will take a very long .me to fill the free energy surface Simulaon .me Energy Collecve variable (CV) R.C. Bernardi, et al., Enhanced sampling techniques in molecular dynamics simulaons of biological systems, Biochim. 26 Biophys. Acta (2014), h_p://dx.doi.org/10.1016/j.bbagen.2014.10.019 Metadynamics - Theory • An external history-dependent bias poten.al which is a func.on of the CVs is added to the Hamiltonian of the system • Poten.al = sum of Gaussians deposited along the system trajectory in the CVs space → discourage the system from revisi.ng configuraons that have already been sampled. Bias potenal: Detachment of nSH2 from helical domain Where: CVs Gaussian ω: energy rate poten.al th σi: the width of the Gaussian for the i CV W: Gaussian height τG: deposi.on stride (pace) of the Gaussians 27 Barducci et. al., John Wiley & Sons, Ltd. WIREs Comput Mol Sci 2011 1 826–843 DOI: 10.1002/wcms.31 Metadynamics - Theory • Gaussian poten.als deposited along the system trajectory in the CVs space • The sum of the Gaussians allows the reconstruc.on of the free energy landscape 28 Barducci et. al., John Wiley & Sons, Ltd. WIREs Comput Mol Sci 2011 1 826–843 DOI: 10.1002/wcms.31 Well-Tempered MetaD Problems of standard MetaD: • It is difficult to decide when to terminate the metaD run • Risk to push the system in regions of configuraonal space that are not physically relevant • The bias poten.al does not converges to a definite value, but rather oscillates around a value with an error The height of the Gaussian is decreased with simulaon .me: Where: W0 is the ini.al Gaussian height ΔT is an input parameter with temperature dimension ΔΤ can be chose to regulate the extent of free energy exploraon •ΔΤ = 0 standard MD •Δ T → ꝏ standard MetaD Bias factor: 29 Parallel-Tempering MetaD 30 Other variants of MetaD Bias-eXchange MetaD • N metadynamics simulaons (replicas) are run on the same system at the same temperature, biasing a different CV in each replica. • Every fixed number of steps, an exchange is aempted between a randomly selected pair of replicas a and b. The probability to accept the exchange is given by a Metropolis rule: Mul7ple Walkers MetaD • Mul.ple simulaons (walkers) of the same system are run in parallel using metadynamics on the same set of collec.ve variables.
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