Jonathan P. K. Doye

The effect of the range of the potential on the structures of clusters

We investigate the structures of clusters bound by the Morse potential by mapping the structure of the global minimum as a function of both cluster size and the range of the pair potential. We consider values of the range parameter appropriate to a loosely bound diatomic molecule (longest), two C60 molecules (shortest), and at regular intervals between these two limits. We have studied all cluster sizes with 25 atoms or less and a selection of sizes containing between 35 and 80 atoms. The effect of decreasing the range of the potential is to destabilize strained structures. For the larger clusters the structure of the global minimum changes from icosahedral to decahedral to face‐centered cubic as the range is decreased. We have also investigated the effects of temperature on the equilibrium structure by performing a model calculation for a 75‐atom cluster.

The double-funnel energy landscape of the 38-atom Lennard-Jones cluster

The 38-atom Lennard-Jones cluster has a paradigmatic double-funnel energy landscape. One funnel ends in the global minimum, a face-centered-cubic (fcc) truncated octahedron. At the bottom of the other funnel is the second lowest energy minimum which is an incomplete Mackay icosahedron. We characterize the energy landscape in two ways. First, from a large sample of minima and transition states we construct a disconnectivity graph showing which minima are connected below certain energy thresholds. Second, we compute the free energy as a function of a bond-order parameter. The free energy profile has two minima, one which corresponds to the fcc funnel and the other which at low temperature corresponds to the icosahedral funnel and at higher temperatures to the liquidlike state. These two approaches show that the greater width of the icosahedral funnel, and the greater structural similarity between the icosahedral structures and those associated with the liquidlike state, are the cause of the smaller free ene...

Evolution of the potential energy surface with size for Lennard-Jones clusters

Disconnectivity graphs are used to characterize the potential energy surfaces of Lennard-Jones clusters containing 13, 19, 31, 38, 55, and 75 atoms. This set includes members which exhibit either one or two “funnels” whose low-energy regions may be dominated by a single deep minimum or contain a number of competing structures. The graphs evolve in size due to these specific size effects and an exponential increase in the number of local minima with the number of atoms. To combat the vast number of minima we investigate the use of monotonic sequence basins as the fundamental topographical unit. Finally, we examine disconnectivity graphs for a transformed energy landscape to explain why the transformation provides a useful approach to the global optimization problem.

Global minima for transition metal clusters described by Sutton–Chen potentials

Using a Monte Carlo minimization approach we report the global minima for metal clusters modelled by the Sutton–Chen family of potentials containing up to 80 atoms. The resulting structures are discussed in the light of both experimental and theoretical data for clusters of the appropriate elements.

On the Biophysics and Kinetics of Toehold-Mediated DNA Strand Displacement

Dynamic DNA nanotechnology often uses toehold-mediated strand displacement for controlling reaction kinetics. Although the dependence of strand displacement kinetics on toehold length has been experimentally characterized and phenomenologically modeled, detailed biophysical understanding has remained elusive. Here, we study strand displacement at multiple levels of detail, using an intuitive model of a random walk on a 1D energy landscape, a secondary structure kinetics model with single base-pair steps and a coarse-grained molecular model that incorporates 3D geometric and steric effects. Further, we experimentally investigate the thermodynamics of three-way branch migration. Two factors explain the dependence of strand displacement kinetics on toehold length: (i) the physical process by which a single step of branch migration occurs is significantly slower than the fraying of a single base pair and (ii) initiating branch migration incurs a thermodynamic penalty, not captured by state-of-the-art nearest neighbor models of DNA, due to the additional overhang it engenders at the junction. Our findings are consistent with previously measured or inferred rates for hybridization, fraying and branch migration, and they provide a biophysical explanation of strand displacement kinetics. Our work paves the way for accurate modeling of strand displacement cascades, which would facilitate the simulation and construction of more complex molecular systems.

Thermodynamics of Global Optimization

Theoretical design of global optimization algorithms can profitably utilize recent statistical mechanical treatments of potential energy surfaces (PESs). Here we analyze a particular method to explain its success in locating global minima on surfaces with a multiple-funnel structure, where trapping in local minima with different morphologies is expected. We find that a key factor in overcoming trapping is the transformation applied to the PES which broadens the thermodynamic transitions. The global minimum then has a significant probability of occupation at temperatures where the free energy barriers between funnels are surmountable.

Tetrahedral global minimum for the 98-atom Lennard-Jones cluster.

An unusual atomic cluster structure corresponding to the global minimum of the 98-atom Lennard-Jones cluster has been found using a variant of the basin-hopping global optimization algorithm. The structure has tetrahedral symmetry and an energy of -543.665 361 epsilon, which is 0.022 404 epsilon lower than the previous lowest-energy minimum. The LJ(98) structure is of particular interest because its tetrahedral symmetry establishes it as one of only three types of exception to the general pattern of icosahedral structural motifs for optimal LJ microclusters. Similar to the other exceptions the global minimum is difficult to find because it is at the bottom of a narrow funnel that only becomes thermodynamically most stable at low temperature.

Thermodynamics and the Global Optimization of Lennard-Jones clusters

Theoretical design of global optimization algorithms can profitably utilize recent statistical mechanical treatments of potential energy surfaces (PES’s). Here we analyze the basin-hopping algorithm to explain its success in locating the global minima of Lennard-Jones (LJ) clusters, even those such as LJ38 for which the PES has a multiple-funnel topography, where trapping in local minima with different morphologies is expected. We find that a key factor in overcoming trapping is the transformation applied to the PES which broadens the thermodynamic transitions. The global minimum then has a significant probability of occupation at temperatures where the free energy barriers between funnels are surmountable.

Network topology of a potential energy landscape: A static scale-free network

Here we analyze the topology of the network formed by the minima and transition states on the potential energy landscape of small clusters. We find that this network has both a small-world and scale-free character. In contrast to other scale-free networks, where the topology results from the dynamics of the network growth, the potential energy landscape is a static entity. Therefore, a fundamentally different organizing principle underlies this behavior: The potential energy landscape is highly heterogeneous with the low-energy minima having large basins of attraction and acting as the highly connected hubs in the network.

Reversible self-assembly of patchy particles into monodisperse icosahedral clusters

We systematically study the design of simple patchy sphere models that reversibly self-assemble into monodisperse icosahedral clusters. We find that the optimal patch width is a compromise between structural specificity (the patches must be narrow enough to energetically select the desired clusters) and kinetic accessibility (they must be sufficiently wide to avoid kinetic traps). Similarly, for good yields the temperature must be low enough for the clusters to be thermodynamically stable, but the clusters must also have enough thermal energy to allow incorrectly formed bonds to be broken. Ordered clusters can form through a number of different dynamic pathways, including direct nucleation and indirect pathways involving large disordered intermediates. The latter pathway is related to a reentrant liquid-to-gas transition that occurs for intermediate patch widths upon lowering the temperature. We also find that the assembly process is robust to inaccurate patch placement up to a certain threshold and that it is possible to replace the five discrete patches with a single ring patch with no significant loss in yield.