Adaptive Brownian Dynamics

Balanced implicit Patankar-Euler methods for positive solutions of stochastic differential equations of biological regulatory systems

Stochastic differential equations (SDEs) are a powerful tool to model fluctuations and uncertainty in complex systems. Although numerical methods have been designed to simulate SDEs effectively, it is still problematic when numerical solutions may be negative, but application problems require positive simulations. To address this issue, we propose balanced implicit Patankar-Euler methods to ensure positive simulations of SDEs. Instead of considering the addition of balanced terms to explicit methods in existing balanced methods, we attempt the deletion of possible negative terms from the explicit methods to maintain positivity of numerical simulations. The designed balanced terms include negative-valued drift terms and potential negative diffusion terms. The proposed method successfully addresses the issue of divisions with very small denominators in our recently designed stochastic Patankar method. Stability analysis shows that the balanced implicit Patankar-Euler method has much better stability properties than our recently designed composite Patankar-Euler method. Four SDE systems are used to examine the effectiveness, accuracy, and convergence properties of balanced implicit Patankar-Euler methods. Numerical results suggest that the proposed balanced implicit Patankar-Euler method is an effective and efficient approach to ensure positive simulations when any appropriate stepsize is used in simulating SDEs of biological regulatory systems.

Simultaneous and independent topological control of identical microparticles in non-periodic energy landscapes

Topological protection ensures stability of information and particle transport against perturbations. We explore experimentally and computationally the topologically protected transport of magnetic colloids above spatially inhomogeneous magnetic patterns, revealing that transport complexity can be encoded in both the driving loop and the pattern. Complex patterns support intricate transport modes when the microparticles are subjected to simple time-periodic loops of a uniform magnetic field. We design a pattern featuring a topological defect that functions as an attractor or a repeller of microparticles, as well as a pattern that directs microparticles along a prescribed complex trajectory. Using simple patterns and complex loops, we simultaneously and independently control the motion of several identical microparticles differing only in their positions above the pattern. Combining complex patterns and complex loops we transport microparticles from unknown locations to predefined positions and then force them to follow arbitrarily complex trajectories concurrently. Our findings pave the way for new avenues in transport control and dynamic self-assembly in colloidal science.

Noether-Constrained Correlations in Equilibrium Liquids

Liquid structure carries deep imprints of an inherent thermal invariance against a spatial transformation of the underlying classical many-body Hamiltonian. At first order in the transformation field Noether's theorem yields the local force balance. Three distinct two-body correlation functions emerge at second order, namely the standard two-body density, the localized force-force correlation function, and the localized force gradient. An exact Noether sum rule interrelates these correlators. Simulations of Lennard-Jones, Yukawa, soft-sphere dipolar, Stockmayer, Gay-Berne and Weeks-Chandler-Andersen liquids, of monatomic water and of a colloidal gel former demonstrate the fundamental role in the characterization of spatial structure.

Perspective: How to overcome dynamical density functional theory

We argue in favour of developing a comprehensive dynamical theory for rationalizing, predicting, designing, and machine learning nonequilibrium phenomena that occur in soft matter. To give guidance for navigating the theoretical and practical challenges that lie ahead, we discuss and exemplify the limitations of dynamical density functional theory (DDFT). Instead of the implied adiabatic sequence of equilibrium states that this approach provides as a makeshift for the true time evolution, we posit that the pending theoretical tasks lie in developing a systematic understanding of the dynamical functional relationships that govern the genuine nonequilibrium physics. While static density functional theory gives a comprehensive account of the equilibrium properties of many-body systems, we argue that power functional theory is the only present contender to shed similar insights into nonequilibrium dynamics, including the recognition and implementation of exact sum rules that result from the Noether theorem. As a demonstration of the power functional point of view, we consider an idealized steady sedimentation flow of the three-dimensional Lennard-Jones fluid and machine-learn the kinematic map from the mean motion to the internal force field. The trained model is capable of both predicting and designing the steady state dynamics universally for various target density modulations. This demonstrates the significant potential of using such techniques in nonequilibrium many-body physics and overcomes both the conceptual constraints of DDFT as well as the limited availability of its analytical functional approximations.

Performance efficient macromolecular mechanics via sub-nanometer shape based coarse graining

Dimensionality reduction via coarse grain modeling is a valuable tool in biomolecular research. For large assemblies, ultra coarse models are often knowledge-based, relying on a priori information to parameterize models thus hindering general predictive capability. Here, we present substantial advances to the shape based coarse graining (SBCG) method, which we refer to as SBCG2. SBCG2 utilizes a revitalized formulation of the topology representing network which makes high-granularity modeling possible, preserving atomistic details that maintain assembly characteristics. Further, we present a method of granularity selection based on charge density Fourier Shell Correlation and have additionally developed a refinement method to optimize, adjust and validate high-granularity models. We demonstrate our approach with the conical HIV-1 capsid and heteromultimeric cofilin-2 bound actin filaments. Our approach is available in the Visual Molecular Dynamics (VMD) software suite, and employs a CHARMM-compatible Hamiltonian that enables high-performance simulation in the GPU-resident NAMD3 molecular dynamics engine.

Reduced-variance orientational distribution functions from torque sampling

We introduce a method to sample the orientational distribution function in computer simulations. The method is based on the exact torque balance equation for classical many-body systems of interacting anisotropic particles in equilibrium. Instead of the traditional counting of events, we reconstruct the orientational distribution function via an orientational integral of the torque acting on the particles. We test the torque sampling method in two- and three-dimensions, using both Langevin dynamics and overdamped Brownian dynamics, and with two interparticle interaction potentials. In all cases the torque sampling method produces profiles of the orientational distribution function with better accuracy than those obtained with the traditional counting method. The accuracy of the torque sampling method is independent of the bin size, and hence it is possible to resolve the orientational distribution function with arbitrarily small angular resolutions.

Inhomogeneous steady shear dynamics of a three-body colloidal gel former

We investigate the stationary flow of a colloidal gel under an inhomogeneous external shear force using adaptive Brownian dynamics simulations. The interparticle forces are derived from the Stillinger-Weber potential, where the three-body term is tuned to enable network formation and gelation in equilibrium. When subjected to the shear force field, the system develops remarkable modulations in the one-body density profile. Depending on the shear magnitude, particles accumulate either in quiescent regions or in the vicinity of maximum net flow, and we deduce this strong non-equilibrium response to be characteristic of the gel state. Studying the components of the internal force parallel and perpendicular to the flow direction reveals that the emerging flow and structure of the stationary state are driven by significant viscous and structural superadiabatic forces. Thereby, the magnitude and nature of the observed non-equilibrium phenomena differ from the corresponding behavior of simple fluids. We demonstrate that a simple power functional theory reproduces accurately the viscous force profile, giving a rationale of the complex dynamical behavior of the system.

Colloidal transport in twisted lattices of optical tweezers

We simulate the transport of colloidal particles driven by a static and homogeneous drift force, and subject to the optical potential created by two lattices of optical tweezers. The lattices of optical tweezers are parallel to each other, shifted, and rotated by a twist angle. Due to a negative interference between the potential of the two lattices, flat channels appear in the total optical potential. At specific twist angles, known as magic angles, the flat channels percolate the entire system and the colloidal particles can then be transported using a weak external drift force. We characterize the transport in both square and hexagonal lattices of twisted optical tweezers.