In experiments and in simulations, the free energy of a state of a system can be determined from the probability that the state is occupied. However, it is often necessary to impose a biasing potential on the system so that high energy states are sampled with a reasonable frequency. The unbiased energy is typically obtained from the data using the weighted histogram analysis method (WHAM). Here we present differential energy surface analysis (DESA), in which the gradient of the energy surface, dE/dx, is extracted from data taken with a series of harmonic biasing potentials. It is shown that DESA produces a maximum likelihood estimate of the folding landscape gradient. DESA is used to analyze extension vs. time data taken as an optical trap is used to unfold a DNA hairpin under a harmonic constraint . It is shown that the energy surface obtained from DESA is indistinguishable from the energy surface obtained when WHAM is applied to the same data. Two criteria are defined which indicate whether the DESA results are self-consistent. It is found that these criteria are satisfied for the experimental data analyzed, confirming that data taken with different constraint origins are sampling the same effective energy surface . The combination of DESA and the optical trap assay in which a structure is disrupted under harmonic constraint facilitates an extremely accurate measurement of the folding energy surface.
In experiments and in simulations, the free energy of a state of a system can be determined from the probability that the state is occupied. However, it is often necessary to impose a biasing potential on the system so that high energy states are sampled with sufficient frequency. The unbiased energy is typically obtained from the data using the weighted histogram analysis method (WHAM). Here we present differential energy surface analysis (DESA), in which the gradient of the energy surface, dE/dx, is extracted from data taken with a series of harmonic biasing potentials. It is shown that DESA produces a maximum likelihood estimate of the folding landscape gradient. DESA is demonstrated by analyzing data from a simulated system as well as data from a single-molecule unfolding experiment in which the end-to-end distance of a DNA hairpin is measured . It is shown that the energy surface obtained from DESA is indistinguishable from the energy surface obtained when WHAM is applied to the same data. Two criteria are defined which indicate whether the DESA results are self-consistent. It is found that these criteria can detect a situation where the energy is not a single-valued function of the measured reaction coordinate. The criteria were found to be satisfied for the experimental data analyzed, confirming that end-to-end distance is a good reaction coordinate for the experimental system . The combination of DESA and the optical trap assay in which a structure is disrupted under harmonic constraint facilitates an extremely accurate measurement of the folding energy surface.
R
a reasonable
sufficient
R
used to analyze extension vs. time data taken as an optical trap is used to unfold a DNA hairpin under a harmonic constraint
demonstrated by analyzing data from a simulated system as well as data from a single-molecule unfolding experiment in which the end-to-end distance of a DNA hairpin is measured
R
are
can detect a situation where the energy is not a single-valued function of the measured reaction coordinate. The criteria were found to be
R
data taken with different constraint origins are sampling the same effective energy surface
end-to-end distance is a good reaction coordinate for the experimental system