This report presents the application of polynomial regression for estimating free energy differences using thermodynamic integration. We employ linear regression to construct a polynomial that optimally fits the thermodynamic integration data, and thus reduces the bias and uncertainty of the resulting free energy estimate. Two test systems with analytical solutions were used to verify the accuracy and precision of the approach. Our results suggest that regression with a high degree of polynomials give the most accurate free energy difference estimates, but often with a slightly larger variance, compared to commonly used quadrature techniques. High degrees of polynomials possess the flexibility to closely fit the thermodynamic integration data but are often sensitive to small changes in data points. To further improve overall accuracy and reduce uncertainty, we also examine the use of Chebyshev nodes to guide the selection of non-equidistant lambda values for the thermodynamic integration scheme. We conclude that polynomial regression with non-equidistant lambda values delivers the most accurate and precise free energy estimates for thermodynamic integration data. Software and documentation is available at http://www.phys.uidaho.edu/ytreberg/software
Deep Dive into Reducing the Bias and Uncertainty of Free Energy Estimates by Using Regression to Fit Thermodynamic Integration Data.
This report presents the application of polynomial regression for estimating free energy differences using thermodynamic integration. We employ linear regression to construct a polynomial that optimally fits the thermodynamic integration data, and thus reduces the bias and uncertainty of the resulting free energy estimate. Two test systems with analytical solutions were used to verify the accuracy and precision of the approach. Our results suggest that regression with a high degree of polynomials give the most accurate free energy difference estimates, but often with a slightly larger variance, compared to commonly used quadrature techniques. High degrees of polynomials possess the flexibility to closely fit the thermodynamic integration data but are often sensitive to small changes in data points. To further improve overall accuracy and reduce uncertainty, we also examine the use of Chebyshev nodes to guide the selection of non-equidistant lambda values for the thermodynamic integrati
Free energy constitutes an important thermodynamic quantity necessary for a complete understanding of most chemical and biochemical processes. Examples such as conformational equilibria and molecular association, partitioning between immiscible liquids, receptor-drug interaction, protein-protein, and protein-DNA association, and protein stability all require the underlying free energy profiles as the prerequisite for a complete comprehension of the intrinsic properties [4,20,21]. Indeed, the grand challenge of molecular modeling is to obtain the microscopic detail that is often inaccessible to conventional experimental techniques. Free energy is typically expressed as the Helmholtz free energy for an isothermal-isochoric system or the Gibbs free energy for an isothermal-isobaric system, respectively [4].
Thermodynamic integration (TI) has been widely employed to calculate free energy differences (∆F ) between two well-defined systems [12,18,23,24,32]. It is a general scheme for the calculation of ∆F between two systems with potential energy functions U 1 and U 0 , respectively. The free energy difference, ∆F = F 1 -F 0 , is the reversible work done when the potential energy function U 0 is continuously and reversibly switched to U 1 , and is defined as
where k B is the Boltzmann constant, T absolute temperature of the system in Kelvin, and the configurational partition function is given by
where R is the full set of configuration coordinates. TI is a method that computes the ∆F between two systems or states of interest by estimating the integral ∆F =
which is equivalent to the reversible work to switch from U 0 → U 1 . The notation • λ represents an ensemble average at a particular value of λ. Switching the system between two potential energies requires a continuously variable energy function U λ such that U λ=0 = U 0 and U λ=1 = U 1 . In addition, the free energy function U λ must be differentiable with respect to λ for 0 ≤ λ ≤ 1 [14].
The relationship of eq. ( 3) is exact, but the integral must be approximated numerically by performing simulation at various discrete values of λ. Typically, these discrete λ values are used to convert the integral to a sum (e.g., using quadrature). If the estimates of • λ include large fluctuations, then it is necessary to perform very long simulations in order to calculate the average value to sufficient statistical accuracy. In addition, the quantity • λ may heavily depend on λ so that a large number of simulations at different λ values is needed in order to estimate the integral with sufficient accuracy.
Typically researchers estimate ∆F with TI utilizing an arithmetic technique such as the trapezoidal or Simpson’s rule. These numerical methods work well if the curvature of the TI data is small. The trapezoidal rule, for example, approximates the area under the curve of a given function with a trapezoid. Thus, ∆F is approximated by summing the area of the trapezoids between λ = 0 and 1. The trapezoidal rule is intrinsically simple to use and possesses the advantage that the sign of the error of the approximation can be determined. The trapezoidal rule will overestimate the integral of a function with a concave-up curve because the trapezoids include all the area under the curve as well as the extension above it. Similarly, an underestimate will likely to occur if the function reveals a concave-down curve because the areas is accounted for under the curve, but not above. However, the error is difficult to estimate if the curve includes an inflection point.
Importantly, the accuracy of ∆F using the trapezoidal rule can only improve by increasing the number of • λ even if the • λ have sufficiently converged. However, such a large number of long equilibrium simulations is not always feasible with limited computational resources.
We previously presented the successful application of polynomial and spline interpolation techniques for ∆F estimates via TI [26]. These techniques demonstrate superior accuracy and precision over trapezoidal quadrature, and give the best estimates of ∆F without demanding additional simulations. However, we also noted the inherent weakness and limitations of the interpolation techniques. The most important weakness is that high degree of interpolating polynomials suffer from Runge’s phenomenon, i.e., the approximation errors escalate rapidly as the degree of interpolating polynomial increases. This phenomenon is attributed to the fact that a data point at or near the middle of the interval gives a large contribution to the coefficients close to the endpoints. As a consequence, there is a tradeoff between having a better fit and obtaining a smooth well-behaved fitting polynomial [8,10].
To alleviate these restrictions on polynomial order, we now introduce the polynomial regression technique for estimating ∆F using TI. Our goal is to reduce the bias and uncertainty in the estimates of ∆F from evaluation of the integral which is present even fo
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