Improving the efficiency of free energy calculations is important for many biological and materials design applications, such as protein-ligand binding affinities in drug design, partitioning between immiscible liquids, and determining molecular association in soft materials. We show that for any pair potential, moderately accurate estimation of the radial distribution function for a solute molecule is sufficient to accurately estimate the statistical variance of a sampling along a free energy pathway. This allows inexpensive analytical identification of low statistical error free energy pathways. We employ a variety of methods to estimate the radial distribution function (RDF) and find that the computationally cheap two-body "dilute gas" limit performs as well or better than 3D-RISM theory and other approximations for identifying low variance free energy pathways. With a RDF estimate in hand, we can search for pairwise interaction potentials that produce low variance. We give an example of a search minimizing statistical variance of solvation free energy over the entire parameter space of a generalized "soft core" potential. The free energy pathway arising from this optimization procedure has lower curvature in the variance and reduces the total variance by at least 50% compared to the traditional soft core solvation pathway. We also demonstrate that this optimized pathway allows free energies to be estimated with fewer intermediate states due to its low curvature. This free energy variance optimization technique is generalizable to solvation in any homogeneous fluid and for any type of pairwise potential and can be performed in minutes to hours, depending on the method used to estimate g(r).

We present a general approach to transform between molecular potential functions during free energy calculations using a variance minimized linear basis functional form. This approach splits the potential energy function into a sum of pairs of basis functions, which depend on coordinates, and 'alchemical' switches, which depend only on the coupling variable. The power of this approach is that, first, the calculation of the coupling parameter dependent terms is removed from inner loop force calculation routines, second, the flexibility in specifying basis functions and alchemical switches allows users to choose transformation pathways that maximize statistical efficiency, and third, it is possible to predict entirely in postprocessing, without any additional energy evaluations, the thermodynamic properties along any alchemical path with moderate overlap from an initial simulation that uses the same basis functions. This allows construction of optimized, minimum variance alchemical switches from a single simulation with fixed basis functions and trial alchemical switching functions. We describe how to construct these linear basis potentials for real molecular systems of different sizes and shapes, considering particularly the problems of eliminating singularities and minimizing variance of particle removal in dense fluids. The statistical error in free energy calculations using the optimized basis functions is lower than standard soft core models, and approach the minimum variance possible over all pair potentials. We recommend an optimized set of basis functions and alchemical switches for standard molecular free energy calculations.

We estimate the global minimum variance path for computing the free energy insertion into or deletion of small molecules from a dense fluid. We perform this optimization over all pair potentials, irrespective of functional form, using functional optimization with a two-body approximation for the radial distribution function. Surprisingly, the optimal pairwise path obtained via this method is almost identical to the path obtained using a optimized generalized "soft core" potential reported by Pham and Shirts [J. Chem. Phys. 135, 034114 (2011)]. We also derive the lowest variance non-pairwise potential path for molecular insertion or deletion and compare its efficiency to the pairwise path. Under certain conditions, non-pairwise pathways can reduce the total variance by up to 60% compared to optimal pairwise pathways. However, optimal non-pairwise pathways do not appear generally feasible for practical free energy calculations because an accurate estimate of the free energy, the parameter that is itself is desired, is required for constructing this non-pairwise path. Additionally, simulations at most intermediate states of these non-pairwise paths have significantly longer correlation times, often exceeding standard simulation lengths for solvation of bulky molecules. The findings suggest that the previously obtained soft core pathway is the lowest variance pathway for molecular insertion or deletion in practice. The findings also demonstrate the utility of functional optimization for determining the efficiency of thermodynamic processes performed with molecular simulation.

Asymmetric cell division, creating sibling cells with distinct developmental potentials, can be manifested in sibling cell size asymmetry. This form of physical asymmetry occurs in several metazoan cells, but the underlying mechanisms and function are incompletely understood. Here we use Drosophila neural stem cells to elucidate the mechanisms involved in physical asymmetry establishment. We show that Myosin relocalizes to the cleavage furrow via two distinct cortical Myosin flows: at anaphase onset, a polarity induced, basally directed Myosin flow clears Myosin from the apical cortex. Subsequently, mitotic spindle cues establish a Myosin gradient at the lateral neuroblast cortex, necessary to trigger an apically directed flow, removing Actomyosin from the basal cortex. On the basis of the data presented here, we propose that spatiotemporally controlled Myosin flows in conjunction with spindle positioning and spindle asymmetry are key determinants for correct cleavage furrow placement and cortical expansion, thereby establishing physical asymmetry.

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