Chapter 2 Methods
2.4 Free Energy Methods
Using the principle of ergodicity, the free energy change associated with moving from stateito statej, ∆Gij, can be approximated from a MD simulation using the
following relationship:
∆Gij =−kBT ln(Pij) (2.26)
wherePij is the ratio of the probability of the system being in statej to the proba-
bility of it being in stateitaken over the length of the simulation; T the temperature of the system; andkBthe Boltzmann constant. For Equation 2.26 to hold it is essen-
tial that both statesiand jare adequately sampled, something which is difficult to achieve in large simulations of systems with complex PEL, as mentioned previously (Section 1.2.4).
In the case of biointerfacial systems, the free energy change of adsorption is of particular interest. In order to calculate this from molecular simulation for systems in which the adsorbate has high affinity for the interface, advanced sampling techniques are often used. For example, Latour and co-workers derived a scheme for calculating the free energy of adsorption of peptides which incorporated both T- REMD–to facilitate sampling of internal peptide conformational space–and a biasing potential–to ensure that the peptide visits both adsorbed and desorbed states [Wang et al. [2008]]. This method has been successfully applied to the adsorption of host- guest TGTG-X-TGTG peptides at a range of interfaces [Vellore et al. [2010]; Snyder et al. [2012]].
A number of alternative free energy approaches exist, two of which have been employed in the work carried out for this thesis (Chapters 3 and 8) and de- scribed in detail below–the Potential of Mean constraint Force (PMF) method and metadynamics (Sections 2.4.1 and 2.4.2, respectively).
2.4.1 Potential of Mean Force
Using a suitable reaction co-ordinate (z′) which connects the two states of interest, the free energy landscape of the system along the same reaction co-ordinate can be calculated using [Trzesniak et al. [2007]]:
∆G(z) = Z z
0
< F(z′)> dz′ (2.27)
where ∆G(z) is the free energy change associated with the transitionz′=0→ z′=z
and< F(z′)>is the average force required to constrain the system atz′. In order
to calculate the free energy of adsorption a small molecule at an interface, a suitable reaction co-ordinate is the vertical distance between the surface and the centre of mass (c.o.m.) of the adsorbate. The PMF method, using this reaction co-ordinate, has in recent years been used to quantify the adsorption of amino-acids (or their analogues) at the Quartz [Notman and Walsh [2009]; Wright and Walsh [2012a]], rutile [Monti and Walsh [2010]] and gold [Hoefling et al. [2010a]] aqueous interfaces. The calculations reported in Chapter 3 were performed using the Gromacs pull code, to calculate the force required to constrain the reaction co-ordinate at each time step. Simulations at each adsorbate-surface separation were run for a minimum of 2 ns or until convergence of force block averages [Hess [2002]] was achieved (up to a maximum of 14 ns). The average force was calculated over the last 1 ns for large adsorbate-surface separations and over the last 2 or 4 ns for adsorbates close to the surface, where the force block averages converged more slowly. Between 50 and 60 MD runs per adsorbate were performed with the separation between the adsorbate and the surface being varied systematically from around 1.0 ˚A to 14 ˚A. A block averaging method [Hess [2002]] was used to estimate errors in the average forces. Average forces were integrated according to Equation 2.27. Errors in the PMF free energy were calculated by propagation of the average force errors in the integration. Whilst by performing non-equilibrium simulations the PMF method ensures that all states of the system that lie along the reaction co-ordinate are adequately sampled, its ability to probe all possible configurations along orthogonal directions on the PEL is not enhanced [Jambeck and Lyubartsev [2013]]. For example, in the case of bio-interfacial PMF calculations, additional measures must be taken to ensure different internal adsorbate and adsorbate-surface geometries are sampled. Small molecules, such as those used in Chapter 6, have few internal degrees of freedom and adequate adsorbate-substrate sampling can be achieved using a small number (2 or 3) of different starting geometries for each restrained simulation. For larger adsorbates, such as peptides, however, the extra computational expense of
ensuring adequate conformational sampling makes the PMF method for calculating the free energy of adsorption prohibitive.
2.4.2 Metadynamics
Metadynamics (metaD) is a method in which the free energy profile of a system along (a) given reaction co-ordinate(s) can be reconstructed from the bias added on-the-fly to a non-equilibrium MD simulation [Laio and Parrinello [2002]]. Briefly, reaction co-ordinates, si, (more commonly, in the context of metaD, referred to as
‘collective variables’) which connect significant minima in the PEL must be chosen. Typically Gaussians of width σi and height w are added with a frequency of 1/τ
to the PEL at the co-ordinates of each collective variable si(X(t)) (where X is the
co-ordinates of the system at time t). The simulation therefore progresses on the time-dependent energy landscape given by the unbiased PEL of the system, U(X), and the metaD bias, V(X,t):
V(X, t) =wΣt′<tΠiexp " −(si(X(t))−si(X(t ′)))2 2σ2 i # . (2.28) As the bias is built up, barriers between low energy states of the system are reduced thus enabling the simulation to escape from deep minima in the unbiased PEL. In the limit t→ ∞:
V(X, t)→ −G(X) (2.29) where G(X,t) is the free energy of the system. In the case of bio-interfacial simu- lation, a suitable collective variable for determining the free energy of adsorption is the distance between the centre of mass (c.o.m.) of an adsorbate and the surface, in the direction of the surface normal. In Chapter 8 the adsorption of a gold binding peptide, AuBP-1 [Hnilova et al. [2008]] onto different crystallographic planes of gold is simulated using metaD. As with PMF calculations, the issues associated with adequately sampling phase-space orthogonal to a collective variable [Jambeck and Lyubartsev [2013]] remain. Hence in the work reported here, the REST method– to augment peptide conformational sampling–has been used in combination with metaD [Camilloni et al. [2008]; Schneider and Ciacchi [2012]]. The same principles lie behind REST metaD and the biased T-REMD method developed by Latour and
co-workers [Wang et al. [2008]; Vellore et al. [2010]; Snyder et al. [2012]]. In the
former, REST metaD, the PEL is biased on-the-fly, while in the latter, expensive umbrella-sampling calculations to determine the form the biasing potential should
take, prior to the T-REMD simulation, are required. ‘Umbrella-sampling’ is a free energy method, analogous to PMF but using a harmonic constraint rather than a fixed restraint on the reaction co-ordinate [Trzesniak et al. [2007]].
When implementing metaD with REST, the metaD bias added to the two replicasi and j involved in a trial exchange must also be included in the potential energy difference, ∆ij, used to determine the probability with which the exchange
is accepted (Equations 2.21):
∆ij ={Ui(Xi) +Vi(Xi) +Uj(Xj) +Vj(Xj)} − {Ui(Xj) +Vi(Xj) +Uj(Xi) +Vj(Xi)}
(2.30) where, U(X) and V(X) refer to the potential energy of the system in the absence of the metaD bias (Equation 2.23) and the metaD bias itself (Equation 2.28), re- spectively. It is noted that there are two principal differences between Hamiltonian replica-exchange metaD (such as REST metaD) and Temperature replica exchange metaD reported by Bussi et al. [Bussi et al. [2006]] in terms of how the two are implemented:
1. The momenta of particles following a successful exchange attempt move are not rescaled;
2. Gaussians of the same height are added to the biasing potential in all replicas. Both of these features arise from the fact that system temperature is invariant in replica space in the former, Hamiltonian replica exchange metaD.
Equilibrium, but not time dependent, properties of a system can be recovered by suitable reweighting of data from non-equilibrium simulations. In general, the unbiased probability of an observable having a specific valueP(Obs) can be derived from the respective biased probability,Pbiased(Obs), using:
P(Obs) = < Pbiased(Obs)exp{
W kT}>
< exp{kTW}> (2.31)
where W is the weight factor. For REST metaD simulations specifically, W is comprised of two components: the time dependent metaD bias and the bias due to the scaling of solute interactions by an effective temperature,Y(X(t)) (derived from Equation 2.23): Y(X(t)) = (1−βj β )Upp(X(t)) + (1− s βj β )Ups(X(t)) (2.32)
The simplest approach, analysis of the reference replica (λ= 0, Equation 2.24) has been adopted only herein–both for REST MD simulations reported in Chapters 5 and 7, and REST metaD simulations detailed in Chapter 8. Incorporating data obtained from all replicas in a REST simulation would be one way to improve statistical sampling in the future.
The factor by which the time dependent metaD bias to the reference replica of a REST metaD simulation must be re-weighted to obtain the unbiased ensemble of states also requires careful consideration. While using V(X(t), t), as defined in Equation 2.28, directly is formally correct, it will exponentially weight frames from the end of the simulation more heavily than those in the initial stages–a consequence ofV(X(t), t) growing continuously with time. Alternative reweighting strategies are discussed in Section 8.3.