Designing self-assembling kinetics with differentiable statistical physics models

2D capsid formation within an oscillatory energy landscape: orderly self-assembly depends on the interplay between a dynamic potential and intrinsic relaxation times

Multiple dissipative self-assembly protocols designed to create novel structures or to reduce kinetic traps have recently emerged. Specifically, temporal oscillations of particle interactions have been shown effective at both aims, but investigations thus far have focused on systems of simple colloids or their binary mixtures. In this work, we expand our understanding of the effect of temporally oscillating interactions to a two-dimensional coarse-grained viral capsid-like model that undergoes a self-limited assembly. This model includes multiple intrinsic relaxation times due to the internal structure of the capsid subunits and, under certain interaction regimes, proceeds via a two-step nucleation mechanism. We find that oscillations much faster than the local intrinsic relaxation times can be described via a time averaged inter-particle potential across a wide range of interaction strengths, while oscillations much slower than these relaxation times result in structures that adapt to the attraction strength of the current half-cycle. Interestingly, oscillation periods similar to these relaxation times shift the interaction window over which orderly assembly occurs by enabling error correction during the half-cycles with weaker attractions. Our results provide fundamental insights to non-equilibrium self-assembly on temporally variant energy landscapes.

Programming patchy particles for materials assembly design

Direct design of complex functional materials would revolutionize technologies ranging from printable organs to novel clean energy devices. However, even incremental steps toward designing functional materials have proven challenging. If the material is constructed from highly complex components, the design space of materials properties rapidly becomes too computationally expensive to search. On the other hand, very simple components such as uniform spherical particles are not powerful enough to capture rich functional behavior. Here, we introduce a differentiable materials design model with components that are simple enough to design yet powerful enough to capture complex materials properties: rigid bodies composed of spherical particles with directional interactions (patchy particles). We showcase the method with self-assembly designs ranging from open lattices to self-limiting clusters, all of which are notoriously challenging design goals to achieve using purely isotropic particles. By directly optimizing over the location and interaction of the patches on patchy particles using gradient descent, we dramatically reduce the computation time for finding the optimal building blocks.

Discovering optimal kinetic pathways for self-assembly using automatic differentiation

Macromolecular complexes are often composed of diverse subunits. The self-assembly of these subunits is inherently nonequilibrium and must avoid kinetic traps to achieve high yield over feasible timescales. We show how the kinetics of self-assembly benefits from diversity in subunits because it generates an expansive parameter space that naturally improves the "expressivity" of self-assembly, much like a deeper neural network. By using automatic differentiation algorithms commonly used in deep learning, we searched the parameter spaces of mass-action kinetic models to identify classes of kinetic protocols that mimic biological solutions for productive self-assembly. Our results reveal how high-yield complexes that easily become kinetically trapped in incomplete intermediates can instead be steered by internal design of rate-constants or external and active control of subunits to efficiently assemble. Internal design of a hierarchy of subunit binding rates generates self-assembly that can robustly avoid kinetic traps for all concentrations and energetics, but it places strict constraints on selection of relative rates. External control via subunit titration is more versatile, avoiding kinetic traps for any system without requiring molecular engineering of binding rates, albeit less efficiently and robustly. We derive theoretical expressions for the timescales of kinetic traps, and we demonstrate our optimization method applies not just for design but inference, extracting intersubunit binding rates from observations of yield-vs.-time for a heterotetramer. Overall, we identify optimal kinetic protocols for self-assembly as a powerful mechanism to achieve efficient and high-yield assembly in synthetic systems whether robustness or ease of "designability" is preferred.

Surrogate Modeling of the Relative Entropy for Inverse Design Using Smolyak Sparse Grids

Relative entropy minimization, a statistical-mechanics approach for finding potential energy functions that produce target structural ensembles, has proven to be a powerful strategy for the inverse design of nanoparticle self-assembly. For a given target structure, the gradient of the relative entropy with respect to the adjustable parameters of the potential energy function is computed by performing a simulation, and then these parameters are updated using iterative gradient-based optimization. Small parameter updates per iteration and many iterations can be required for numerical stability, but this incurs considerable computational expense because a new simulation must be performed to reevaluate the gradient at each iteration. Here, we investigate the use of surrogate modeling to decouple the process of minimizing the relative entropy from the computationally demanding process of determining its gradient. We approximate the relative-entropy gradient using Chebyshev polynomial interpolation on Smolyak sparse grids. Our approach potentially increases the robustness and computational efficiency of using the relative entropy for inverse design, primarily for physically informed potential energy functions that have a small number of adjustable parameters.

Adaptive nonequilibrium design of actin-based metamaterials: Fundamental and practical limits of control

The adaptive and surprising emergent properties of biological materials self-assembled in far-from-equilibrium environments serve as an inspiration for efforts to design nanomaterials. In particular, controlling the conditions of self-assembly can modulate material properties, but there is no systematic understanding of either how to parameterize external control or how controllable a given material can be. Here, we demonstrate that branched actin networks can be encoded with metamaterial properties by dynamically controlling the applied force under which they grow and that the protocols can be selected using multi-task reinforcement learning. These actin networks have tunable responses over a large dynamic range depending on the chosen external protocol, providing a pathway to encoding "memory" within these structures. Interestingly, we obtain a bound that relates the dissipation rate and the rate of "encoding" that gives insight into the constraints on control-both physical and information theoretical. Taken together, these results emphasize the utility and necessity of nonequilibrium control for designing self-assembled nanostructures.

Non-reciprocity across scales in active mixtures

In active matter, particles typically experience mediated interactions, which are not constrained by Newton's third law and are therefore generically non-reciprocal. Non-reciprocity leads to a rich set of emerging behaviors that are hard to account for starting from the microscopic scale, due to the absence of a generic theoretical framework out of equilibrium. Here we consider bacterial mixtures that interact via mediated, non-reciprocal interactions (NRI) like quorum-sensing and chemotaxis. By explicitly relating microscopic and macroscopic dynamics, we show that, under conditions that we derive explicitly, non-reciprocity may fade upon coarse-graining, leading to large-scale equilibrium descriptions. In turn, this allows us to account quantitatively, and without fitting parameters, for the rich behaviors observed in microscopic simulations including phase separation, demixing, and multi-phase coexistence. We also derive the condition under which non-reciprocity survives coarse-graining, leading to a wealth of dynamical patterns. Again, our analytical approach allows us to predict the phase diagram of the system starting from its microscopic description. All in all, our work demonstrates that the fate of non-reciprocity across scales is a subtle and important question.

Nonequilibrium design strategies for functional colloidal assemblies

We use a nonequilibrium variational principle to optimize the steady-state, shear-induced interconversion of self-assembled nanoclusters of DNA-coated colloids. Employing this principle within a stochastic optimization algorithm allows us to identify design strategies for functional materials. We find that far-from-equilibrium shear flow can significantly enhance the flux between specific colloidal states by decoupling trade-offs between stability and reactivity required by systems in equilibrium. For isolated nanoclusters, we find nonequilibrium strategies for amplifying transition rates by coupling a given reaction coordinate to the background shear flow. We also find that shear flow can be made to selectively break detailed balance and maximize probability currents by coupling orientational degrees of freedom to conformational transitions. For a microphase consisting of many nanoclusters, we study the flux of colloids hopping between clusters. We find that a shear flow can amplify the flux without a proportional compromise on the microphase structure. This approach provides a general means of uncovering design principles for nanoscale, autonomous, functional materials driven far from equilibrium.

Exploring the free-energy landscape of a rotating superfluid

The equilibrium state of a superfluid in a rotating cylindrical vessel is a vortex crystal-an array of vortex lines, which is stationary in the rotating frame. Experimental realizations of this behavior typically show a sequence of transient states before the free-energy-minimizing configuration is reached. Motivated by these observations, we construct a new method for a systematic exploration of the free-energy landscape via gradient-based optimization of a scalar loss function. Our approach is inspired by the pioneering numerical work of Campbell and Ziff [Phys. Rev. B. 20, 1886 (1979)] and makes use of automatic differentiation, which crucially allows us to include entire solution trajectories in the loss. We first use the method to converge thousands of low free-energy relative equilibria in the unbounded domain for vortex numbers in the range 10≤N≤30, which reveals an extremely dense set of mostly saddle-like solutions. As part of this search, we discover new continuous families of relative equilibria, which are often global minimizers of free energy. These continuous families all consist of crystals arranged in a double-ring configuration, and we assess which state from the family is most likely to be observed experimentally by computing energy-minimizing pathways from nearby local minima-identifying a common entry point into the family. The continuous families become discrete sets of equal-energy solutions when the wall is introduced in the problem. Finally, we develop an approach to compute homoclinic orbits and use it to examine the dynamics in the vicinity of the minimizing state by converging connections for low-energy saddles.

Discovering optimal kinetic pathways for self-assembly using automatic differentiation

During self-assembly of macromolecules ranging from ribosomes to viral capsids, the formation of long-lived intermediates or kinetic traps can dramatically reduce yield of the functional products. Understanding biological mechanisms for avoiding traps and efficiently assembling is essential for designing synthetic assembly systems, but learning optimal solutions requires numerical searches in high-dimensional parameter spaces. Here, we exploit powerful automatic differentiation algorithms commonly employed by deep learning frameworks to optimize physical models of reversible self-assembly, discovering diverse solutions in the space of rate constants for 3-7 subunit complexes. We define two biologically-inspired protocols that prevent kinetic trapping through either internal design of subunit binding kinetics or external design of subunit titration in time. Our third protocol acts to recycle intermediates, mimicking energy-consuming enzymes. Preventative solutions via interface design are the most efficient and scale better with more subunits, but external control via titration or recycling are effective even for poorly evolved binding kinetics. Whilst all protocols can produce good solutions, diverse subunits always helps; these complexes access more efficient solutions when following external control protocols, and are simpler to design for internal control, as molecular interfaces do not need modification during assembly given sufficient variation in dimerization rates. Our results identify universal scaling in the cost of kinetic trapping, and provide multiple strategies for eliminating trapping and maximizing assembly yield across large parameter spaces.

Active foam: the adaptive mechanics of 2D air-liquid foam under cyclic inflation

Foam is a canonical example of disordered soft matter where local force balance leads to the competition of many metastable configurations. We present an experimental and theoretical framework for "active foam" where an individual voxel inflates and deflates periodically. Local periodic activity leads to irreversible and reversible T1 transitions throughout the foam, eventually reaching a reversible limit cycle. Individual vertices displace outwards and subsequently return back to their approximate original radial position; this radial displacement follows an inverse law. Surprisingly, each return trajectory does not retrace its outbound path but encloses a finite area, with a clockwise (CW) or counterclockwise (CCW) direction, which we define as a local swirl. These swirls form coherent patterns spanning the scale of the material. Using a dynamical model, we demonstrate that swirl arises from disorder in the local micro-structure. We demonstrate that disorder and strain-rate control a crossover between cooperation and competition between swirls in adjacent vertices. Over 5-10 cycles, the region around the active voxel structurally adapts from a higher-energy metastable state to a lower-energy state, locally ordering and stiffening the structure. The coherent domains of CW/CCW swirl become smaller as the system stabilizes, indicative of a process similar to the Hall-Petch effect. Finally, we introduce a statistical model that evolves edge lengths with a set of rules to explore how this class of materials adapts as a function of initial structure. Adding activity to foam couples structural disorder and adaptive dynamics to encourage the development of a new class of abiotic, cellularized active matter.

Essay: Collections of Deformable Particles Present Exciting Challenges for Soft Matter and Biological Physics

The field of soft matter physics has expanded rapidly over the past several decades, as physicists realize that a broad set of materials and systems are amenable to a physical understanding based on the interplay of entropy, elasticity, and geometry. The fields of biological physics and the physics of living systems have similarly emerged as bona fide independent areas of physics in part because tools from molecular and cell biology and optical physics allow scientists to make new quantitative measurements to test physical principles in living systems. This Essay will highlight two exciting future challenges I see at the intersection of these two fields: characterizing emergent behavior and harnessing actuation in highly deformable active objects. I will attempt to show how this topic is a natural extension of older and more recent discoveries and why I think it is likely to unfurl into a wide range of projects that can transform both fields. Progress in this area will enable new platforms for creating adaptive smart materials that can execute large-scale changes in shape in response to stimuli and improve our understanding of biological function, potentially allowing us to identify new targets for fighting disease. Part of a series of Essays which concisely present author visions for the future of their field.

Deep coarse-grained potentials via relative entropy minimization

Neural network (NN) potentials are a natural choice for coarse-grained (CG) models. Their many-body capacity allows highly accurate approximations of the potential of mean force, promising CG simulations of unprecedented accuracy. CG NN potentials trained bottom-up via force matching (FM), however, suffer from finite data effects: They rely on prior potentials for physically sound predictions outside the training data domain, and the corresponding free energy surface is sensitive to errors in the transition regions. The standard alternative to FM for classical potentials is relative entropy (RE) minimization, which has not yet been applied to NN potentials. In this work, we demonstrate, for benchmark problems of liquid water and alanine dipeptide, that RE training is more data efficient, due to accessing the CG distribution during training, resulting in improved free energy surfaces and reduced sensitivity to prior potentials. In addition, RE learns to correct time integration errors, allowing larger time steps in CG molecular dynamics simulation, while maintaining accuracy. Thus, our findings support the use of training objectives beyond FM, as a promising direction for improving CG NN potential's accuracy and reliability.

Algorithmic Differentiation for Automated Modeling of Machine Learned Force Fields

Reconstructing force fields (FFs) from atomistic simulation data is a challenge since accurate data can be highly expensive. Here, machine learning (ML) models can help to be data economic as they can be successfully constrained using the underlying symmetry and conservation laws of physics. However, so far, every descriptor newly proposed for an ML model has required a cumbersome and mathematically tedious remodeling. We therefore propose using modern techniques from algorithmic differentiation within the ML modeling process, effectively enabling the usage of novel descriptors or models fully automatically at an order of magnitude higher computational efficiency. This paradigmatic approach enables not only a versatile usage of novel representations and the efficient computation of larger systems─all of high value to the FF community─but also the simple inclusion of further physical knowledge, such as higher-order information (e.g., Hessians, more complex partial differential equations constraints etc.), even beyond the presented FF domain.

Normalizing flows for atomic solids

We present a machine-learning approach, based on normalizing flows, for modelling atomic solids. Our model transforms an analytically tractable base distribution into the target solid without requiring ground-truth samples for training. We report Helmholtz free energy estimates for cubic and hexagonal ice modelled as monatomic water as well as for a truncated and shifted Lennard-Jones system, and find them to be in excellent agreement with literature values and with estimates from established baseline methods. We further investigate structural properties and show that the model samples are nearly indistinguishable from the ones obtained with molecular dynamics. Our results thus demonstrate that normalizing flows can provide high-quality samples and free energy estimates without the need for multi-staging.

Correlation Tracking: Using simulations to interpolate highly correlated particle tracks

Despite significant advances in particle imaging technologies over the past two decades, few advances have been made in particle tracking, i.e., linking individual particle positions across time series data. The state-of-the-art tracking algorithm is highly effective for systems in which the particles behave mostly independently. However, these algorithms become inaccurate when particle motion is highly correlated, such as in dense or strongly interacting systems. Accurate particle tracking is essential in the study of the physics of dense colloids, such as the study of dislocation formation, nucleation, and shear transformations. Here, we present a method for particle tracking that incorporates information about the correlated motion of the particles. We demonstrate significant improvement over the state-of-the-art tracking algorithm in simulated data on highly correlated systems.

Large self-assembled clathrin lattices spontaneously disassemble without sufficient adaptor proteins

Clathrin-coated structures must assemble on cell membranes to internalize receptors, with the clathrin protein only linked to the membrane via adaptor proteins. These structures can grow surprisingly large, containing over 20 clathrin, yet they often fail to form productive vesicles, instead aborting and disassembling. We show that clathrin structures of this size can both form and disassemble spontaneously when adaptor protein availability is low, despite high abundance of clathrin. Here, we combine recent in vitro kinetic measurements with microscopic reaction-diffusion simulations and theory to differentiate mechanisms of stable vs unstable clathrin assembly on membranes. While in vitro conditions drive assembly of robust, stable lattices, we show that concentrations, geometry, and dimensional reduction in physiologic-like conditions do not support nucleation if only the key adaptor AP-2 is included, due to its insufficient abundance. Nucleation requires a stoichiometry of adaptor to clathrin that exceeds 1:1, meaning additional adaptor types are necessary to form lattices successfully and efficiently. We show that the critical nucleus contains ~25 clathrin, remarkably similar to sizes of the transient and abortive structures observed in vivo. Lastly, we quantify the cost of bending the membrane under our curved clathrin lattices using a continuum membrane model. We find that the cost of bending the membrane could be largely offset by the energetic benefit of forming curved rather than flat structures, with numbers comparable to experiments. Our model predicts how adaptor density can tune clathrin-coated structures from the transient to the stable, showing that active energy consumption is therefore not required for lattice disassembly or remodeling during growth, which is a critical advance towards predicting productive vesicle formation.

Automatic differentiation to simultaneously identify nonlinear dynamics and extract noise probability distributions from data

The sparse identification of nonlinear dynamics (SINDy) is a regression framework for the discovery of parsimonious dynamic models and governing equations from time-series data. As with all system identification methods, noisy measurements compromise the accuracy and robustness of the model discovery procedure. In this work we develop a variant of the SINDy algorithm that integrates automatic differentiation and recent time-stepping constrained motivated by Rudy et al (2019 J. Computat. Phys. 396 483–506) for simultaneously (1) denoising the data, (2) learning and parametrizing the noise probability distribution, and (3) identifying the underlying parsimonious dynamical system responsible for generating the time-series data. Thus within an integrated optimization framework, noise can be separated from signal, resulting in an architecture that is approximately twice as robust to noise as state-of-the-art methods, handling as much as 40% noise on a given time-series signal and explicitly parametrizing the noise probability distribution. We demonstrate this approach on several numerical examples, from Lotka-Volterra models to the spatio-temporal Lorenz 96 model. Further, we show the method can learn a diversity of probability distributions for the measurement noise, including Gaussian, uniform, Gamma, and Rayleigh distributions.

The generalized continuous multiple step (GCMS) potential: model systems and benchmarks

The generalized continuous multiple step (GCMS) potential is presented in this work. Its flexible form allows forrepulsiveand/orattractivecontributions to be encoded through adjustable energy and length scales. The GCMS interaction provides a continuous representation of square-well, square-shoulder potentials and their variants for implementation in computer simulations. A continuous and differentiable energy representation is required to derive forces in conventional simulation algorithms. Molecular dynamics simulations are of particular interest when considering the dynamic properties of a system. The GCMS potential can mimic other interactions with a judicious choice of parameters due to the versatile sigmoid form. In this study, our benchmarks for the GCMS representation include triangular, Yukawa, Franzese, and Lennard-Jones potentials. Comparisons made with published data on volumetric phase diagrams, liquid structure, and diffusivity from model systems are in excellent agreement.

Learning neural network potentials from experimental data via Differentiable Trajectory Reweighting

In molecular dynamics (MD), neural network (NN) potentials trained bottom-up on quantum mechanical data have seen tremendous success recently. Top-down approaches that learn NN potentials directly from experimental data have received less attention, typically facing numerical and computational challenges when backpropagating through MD simulations. We present the Differentiable Trajectory Reweighting (DiffTRe) method, which bypasses differentiation through the MD simulation for time-independent observables. Leveraging thermodynamic perturbation theory, we avoid exploding gradients and achieve around 2 orders of magnitude speed-up in gradient computation for top-down learning. We show effectiveness of DiffTRe in learning NN potentials for an atomistic model of diamond and a coarse-grained model of water based on diverse experimental observables including thermodynamic, structural and mechanical properties. Importantly, DiffTRe also generalizes bottom-up structural coarse-graining methods such as iterative Boltzmann inversion to arbitrary potentials. The presented method constitutes an important milestone towards enriching NN potentials with experimental data, particularly when accurate bottom-up data is unavailable.

Probing the theoretical and computational limits of dissipative design

Self-assembly, the process by which interacting components form well-defined and often intricate structures, is typically thought of as a spontaneous process arising from equilibrium dynamics. When a system is driven by external nonequilibrium forces, states statistically inaccessible to the equilibrium dynamics can arise, a process sometimes termed direct self-assembly. However, if we fix a given target state and a set of external control variables, it is not well-understood (i) how to design a protocol to drive the system toward the desired state nor (ii) the cost of persistently perturbing the stationary distribution. In this work, we derive a bound that relates the proximity to the chosen target with the dissipation associated with the external drive, showing that high-dimensional external control can guide systems toward target distribution but with an inevitable cost. Remarkably, the bound holds arbitrarily far from equilibrium. Second, we investigate the performance of deep reinforcement learning algorithms and provide evidence for the realizability of complex protocols that stabilize otherwise inaccessible states of matter.

Thermodynamic stability versus kinetic accessibility: Pareto fronts for programmable self-assembly

A challenge in designing self-assembling building blocks is to ensure the target state is both thermodynamically stable and kinetically accessible. These two objectives are known to be typically in competition, but it is not known how to simultaneously optimize them. We consider this problem through the lens of multi-objective optimization theory: we develop a genetic algorithm to compute the Pareto fronts characterizing the tradeoff between equilibrium probability and folding rate, for a model system of small polymers of colloids with tunable short-ranged interaction energies. We use a coarse-grained model for the particles' dynamics that allows us to efficiently search over parameters, for systems small enough to be enumerated. For most target states there is a tradeoff when the number of types of particles is small, with medium-weak bonds favouring fast folding, and strong bonds favouring high equilibrium probability. The tradeoff disappears when the number of particle types reaches a value m*, that is usually much less than the total number of particles. This general approach of computing Pareto fronts allows one to identify the minimum number of design parameters to avoid a thermodynamic-kinetic tradeoff. However, we argue, by contrasting our coarse-grained model's predictions with those of Brownian dynamics simulations, that particles with short-ranged isotropic interactions should generically have a tradeoff, and avoiding it in larger systems will require orientation-dependent interactions.