Free Energy Calculations^[1]^[2]

Equilibrium Methods (FEP, TI)
Nonequilibrium Methods

Density of States

$\Omega$ = density of states
$1/h^{3N}$ = factor of proportionality (necessary to retrieve correct correspondence with high-temperature quantum-mechanical prediction)

Calculations

In canonical ensemble, $A=-\beta \ln {Q(N,V,T)}$

Ergodicity

The ergodic hypothesis allows generating trajectories (i.e. MD, MC) to sample from phase-space to compute ensemble averages.
The ensemble average of the property $f$ of a system is:

$\left\langle f\right\rangle =\int fP(\mathbf {x} ,\mathbf {p} _{x})d\mathbf {x} d\mathbf {p} _{x}$

The ergodic hypothesis states that

$\left\langle f\right\rangle =\lim _{T\rightarrow \infty }{\frac {1}{T}}\int _{0}^{T}f[\mathbf {q} (t),\mathbf {p} (t)]dt$ for any $\mathbf {q_{0}} =\mathbf {q} (0),\mathbf {p_{0}} =\mathbf {p} (0)$

Quasi-nonergodic - a system that does not explore phase space over time.
- high energy barriers separating volumes of phase space
- volume in phase space diffuses very slowly
Proof of Quasi-ergodic hypothesis^[3]
Proof of ergodic theorem^[4]

Free Energy Perturbation

Perturbation Formalism

We want to calculate the free energy difference between state 0 and state 1 of Hamiltonians ${\mathcal {H}}_{0}(\mathbf {x} ,\mathbf {p} _{x})$ and ${\mathcal {H}}_{1}(\mathbf {x} ,\mathbf {p} _{x})$

The difference in Helmholtz free energy between is $\Delta A=-{\frac {1}{\beta }}\ln {\frac {Q_{1}}{Q_{0}}}$ where $\beta =(k_{B}T)^{-1}$

$Q_{0}$ and $Q_{1}$ are their corresponding partition functions $Q={\frac {1}{h^{3N}N!}}\iint \exp[-\beta {\mathcal {H}}]d\mathbf {x} d\mathbf {p} _{x}$

$\Delta A=-{\frac {1}{\beta }}\ln {\frac {\iint \exp[-\beta {\mathcal {H}}_{1}(\mathbf {x} ,\mathbf {p} _{x})]d\mathbf {x} d\mathbf {p} _{x}}{\iint \exp[-\beta {\mathcal {H}}_{0}(\mathbf {x} ,\mathbf {p} _{x})]d\mathbf {x} d\mathbf {p} _{x}}}$

$\Delta A=-{\frac {1}{\beta }}\ln {\frac {\iint \exp[-\beta \Delta {\mathcal {H}}(\mathbf {x} ,\mathbf {p} _{x})]\exp[-\beta {\mathcal {H}}_{0}(\mathbf {x} ,\mathbf {p} _{x})]d\mathbf {x} d\mathbf {p} _{x}}{\iint \exp[-\beta {\mathcal {H_{0}}}]d\mathbf {x} d\mathbf {p} _{x}}}$

$\Delta A=-{\frac {1}{\beta }}\ln \iint \exp[-\beta \Delta {\mathcal {H}}(\mathbf {x} ,\mathbf {p} _{x})]P_{0}(\mathbf {x} ,\mathbf {p} _{x})d\mathbf {x} d\mathbf {p} _{x}$ $\Delta A=-{\frac {1}{\beta }}\ln \iint \exp[-\beta \Delta {\mathcal {H}}(\mathbf {x} ,\mathbf {p} _{x})]P_{0}(\mathbf {x} ,\mathbf {p} _{x})d\mathbf {x} d\mathbf {p} _{x}$
- since $P_{0}(\mathbf {x} ,\mathbf {p} _{x})={\frac {\exp[-\beta \Delta {\mathcal {H}}_{0}(\mathbf {x} ,\mathbf {p} _{x})]}{\iint \exp[-\beta {\mathcal {H}}_{0}(\mathbf {x} ,\mathbf {p} _{x})]d\mathbf {x} d\mathbf {p} _{x}}}$

Finally, $\Delta A=-{\frac {1}{\beta }}\ln \left\langle \exp[-\beta \Delta {\mathcal {H}}(\mathbf {x} ,\mathbf {p} _{x})]\right\rangle _{0}$

For systems of with the same masses, $\Delta A=-{\frac {1}{\beta }}\ln \left\langle \exp[-\beta \Delta U]\right\rangle _{0}$

We can also swap states 0 and 1 and use $\Delta A=-{\frac {1}{\beta }}\ln \left\langle \exp[-\beta \Delta U]\right\rangle _{1}$ $\Delta A=-{\frac {1}{\beta }}\ln \left\langle \exp[-\beta \Delta U]\right\rangle _{1}$
- Although forward and backward calculations are formally equilvalent, they may have different convergence properties

Forward and backward calculations can be combined (BAR).

Stratification

P( $\Omega$ ) is a probability distribution over the domain $\Omega$
Partition $\Omega$ in to disjoint regions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega_i\subset\Omega} so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigcup_{i\epsilon I}\Omega_i=\Omega}
Example
- Consider a transformation of state 0 to state 1 described by parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda}
- Suppose states 0 and 1 are separated by a high-energy barrier
- Partition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} in to a number of smaller intervals
- P(Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} ) can be recovered with great savings in computer time
Decreases variance of ensemble average
Provides solution for quasi-nonergodic systems

Importance sampling

Sample regions in phase space that are important for estimating quantity of interest more frequently
Results of the simulation need to be weighted to ensure that estimator is unbiased
Example
- Transformation from state 0 to state 1 along Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda}
- Define P(Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} ) to be the true distribution
- Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P'(\lambda)=P(\lambda)\exp[\beta\eta(\lambda)]} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta(\lambda)} is the weighting factor
- Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta A =-\beta^{-1} \ln \frac{P(\lambda_1)}{P(\lambda_0)}=-\beta^{-1} \ln \frac{P'(\lambda_1)}{P'(\lambda_0)}+\eta(\lambda_1)-\eta(\lambda_0)}

Order Parameter

Coupling parameters, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} , that are used to describe transformations from the reference system to the target one.
Reaction coordinate/path - an order parameters that corresponds to the path along which the transformation takes place in nature
Examples
- Simple/coupled torsional degrees of freedom
- End-to-end reaction coordinate (for protein folding)
- Mutagenesis of atoms (relative free energy)
- Distance separating chemical species (PMF)
- Annihilation of nonbonded interactions (solvation)
Choice of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} significantly affects efficiency, accuracy of calculations
- Smooth energy landscapes require less intermediate states than rough ones

Topology

Single Topology paradigm - a common topology is used for initial and final states
- Missing atoms are treated as vanishing particles by setting nonbonded parameters to 0.
- Bonded terms
  - If chemical bonds are not strongly deformed between initial and target states, bonded terms are often not considered
  - However, FEP and TI can simulations can be used as well.
- Examples of systems that can use single topology

^[5]

Dual Topology paradigm -
- States 0 and 1 contain different topologies
- Atoms not common to 0 and 1 never interact during the simulation.
- Bonded parameters are not perturbed

- ^[6]

Thermodynamic Cycle

Hydration free energy of a small solute
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta A_{hydration} = \Delta A_0 - \Delta A_1}

Binding free energy of ligand to protein
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta A_{binding} = \Delta A_0 - \Delta A_1}

Relative binding free energy of ligand to protein
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \Delta A_{AB} = \Delta A_{bind\!-\!B} - \Delta A_{bind\!-\!A} = \Delta A_1 - \Delta A_0}

Application of FEP

Enthalpy and entropy can be computed based on their thermodynamic relationship using finite difference.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta S = \partial \Delta A / \partial T}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta U = \partial \beta \Delta A / \partial \beta}

Thermodynamic Integration

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \xi} is a reaction coordinate, the free energy is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(\xi)=-k_BT\ln P(\xi)}
For a sufficiently smooth function, free energy difference can be written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(\xi_1)-A(\xi_0)=\int_{\xi_0}^{\xi_1}dA/d\xi \, d\xi} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dA}{d\xi} = \left \langle \frac{\partial \mathcal{H}}{\partial\xi} \right \rangle_\xi}
Order parameters can be torsion angle, bond lengths/angles between atoms or groups of atoms, hydration number, gyration radius

Constrained Simulations

The Hamiltonian is supplemented by a Langrange multiplier

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}^\lambda \equiv \mathcal{H} + \lambda (\xi - \xi(\mathbf{x})) = K + U + \lambda (\xi - \xi(\mathbf{x}))}

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \xi(\mathbf{x}) } is constant, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ddot{\xi} = 0} we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda = Z_{\xi}^{-1} \left ( \sum_i \frac{1}{m_i} \frac{\partial \xi}{\partial x_i} \frac{\partial U}{\partial x_i} - \dot{\mathbf{x}}^t \cdot \mathbf{H} \cdot \dot{\mathbf{x}} \right ) } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{\xi} = \sum_i \frac{1}{m_i} \left ( \frac{\partial \xi}{\partial x_i} \right )^2} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{H}_{ij}=\frac{\partial ^2 \xi}{\partial x_i \partial x_j}} is the hessian of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \xi}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(\xi) = \int \langle \lambda \rangle_{\xi,\dot{\xi}}d\xi - k_bT\ln \langle Z_{\xi}^{-1/2} \rangle }

Adaptive Biasing Force Method^[7]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dA}{d\xi}=- \left \langle \frac{d}{dt}\left ( m_{\xi}\frac{d\xi}{dt} \right ) \right \rangle} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{\xi} = Z_{\xi}^{-1}}
Using velocity verlet integrator where:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(t+dt/2)=v(t-dt/2)+dt \; a(t)}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t+dt/2)=x(t)+dt/2 \; v(t+dt/2)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt}\left ( m_{\xi}\frac{d\xi}{dt} \right ) = \frac{1}{2} \left ( \frac{p_\xi^+ (t+dt)- p_\xi^+(t)}{dt} + \frac{p_\xi^- (t+dt)- p_\xi^-(t)}{dt}\right ) }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_\xi^+ (t) \equiv m_\xi (t) grad(\xi(t)) \cdot \left [v(t+dt/2)-\frac{dt}{6}a(t+dt) \right ]} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_\xi^- (t) \equiv m_\xi (t) grad(\xi(t)) \cdot \left [v(t-dt/2)-\frac{dt}{6}a(t-dt) \right ]}

Ramping function is used to prevent nonequilibrium effects and systematic bias in the beginning of the simulation.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R(n)=\begin{cases} n/N_0 & \textrm{if } \; n \leq N_0 \\ 1 & \textrm{if } \; n > N_0 \\ \textrm {with } \; N_0 \cong 100 \end{cases}}

Examples

Deca-L-alanine

ABF was used to probe the reversible unfolding of deca-L-alanine in vacuo
The reaction coordinate is the distance separating first and last alpha carbon from 12-32 Angstrom
- 5ns MD trajectory
Minimum at around 14 Angstrom
Free energy profile essentially identical to a reversible, 20-ns steered MD simulation
Park et al.^[8]

Glycophorin A

ABF was used to model the helix-helix association of Glycophorin A
Difficulty of this simulation is that once the two helices are close to each other, it is very difficult to slide one with respect to the other or change relative orientation due to steric clashes.
Constrained sampling would keep positions in local minimum in phase-space.

References

↑ Frenkel, D.; Smit, B., Understanding molecular simulation: from algorithms to applications. Academic: San Diego, Calif. London, 2002.
↑ Chipot, C.; Pohorille, A., Free Energy Calculations - Theory and Applications in Chemistry and Biology. Springer-Verlag: Berlin Heidelberg, 2007; Vol. 86.
↑ http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076162
↑ http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076138
↑ Jiao, D.; Zhang, J.; Duke, R. E.; Li, G.; Schnieders, M. J.; Ren, R. J. Comput. Chem. 2009, 30, 1701– 1711.
↑ http://www.ks.uiuc.edu/Research/namd/2.6/ug/node36.html#fig:dual_top
↑ E. Darve and A. Pohorille, J. Chem. Phys. 115, 9169 (2001).
↑ Park, S.; Khalili-Araghi, F.; Tajkhorshid, E.; Schulten, K., Free energy ccalculation from steered molecular dynamics simulatiosn using Jarzynski's equality, J. Chem. Phys. 2003, 119, 3559-3566.

[chipot2007-1] Frenkel, D.; Smit, B., Understanding molecular simulation: from algorithms to applications. Academic: San Diego, Calif. London, 2002.

[frenkel2002-2] Chipot, C.; Pohorille, A., Free Energy Calculations - Theory and Applications in Chemistry and Biology. Springer-Verlag: Berlin Heidelberg, 2007; Vol. 86.

[3] ttp://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076162

[4] ttp://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076138

[5] Jiao, D.; Zhang, J.; Duke, R. E.; Li, G.; Schnieders, M. J.; Ren, R. J. Comput. Chem. 2009, 30, 1701– 1711.

[6] ttp://www.ks.uiuc.edu/Research/namd/2.6/ug/node36.html#fig:dual_top

[7] E. Darve and A. Pohorille, J. Chem. Phys. 115, 9169 (2001).

[8] Park, S.; Khalili-Araghi, F.; Tajkhorshid, E.; Schulten, K., Free energy ccalculation from steered molecular dynamics simulatiosn using Jarzynski's equality, J. Chem. Phys. 2003, 119, 3559-3566.

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Ammm:Free energy simulations

Contents

Free Energy Calculations^[1]^[2]

Density of States

Ergodicity

Free Energy Perturbation

Perturbation Formalism

Stratification

Importance sampling

Order Parameter

Topology

Thermodynamic Cycle

Application of FEP

Thermodynamic Integration

Constrained Simulations

Adaptive Biasing Force Method^[7]

Examples

Deca-L-alanine

Glycophorin A

References

Navigation menu

Ammm:Free energy simulations

Free Energy Calculations[1][2]

Density of States

Ergodicity

Free Energy Perturbation

Perturbation Formalism

Stratification

Importance sampling

Order Parameter

Topology

Thermodynamic Cycle

Application of FEP

Thermodynamic Integration

Constrained Simulations

Adaptive Biasing Force Method[7]

Examples

Deca-L-alanine

Glycophorin A

References

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Free Energy Calculations^[1]^[2]

Adaptive Biasing Force Method^[7]