General Purpose Inverse Design of Heterogeneous Finite-Sized Assemblies

TL;DR

The paper addresses programming heterogeneous, self-assembling building blocks to achieve target equilibrium yields by inverting a closed-form analytical yield calculation. It develops an end-to-end differentiable framework that computes partition functions $Z_s$ and maps them to equilibrium concentrations $c_s$ via a self-consistent system, enabling gradient-based optimization with respect to control parameters $\bm{\theta}$ such as temperatures, interaction strengths, and monomer concentrations. Key innovations include differentiable implicit differentiation to backprop through the fixed-point equations, multi-ensemble optimization for temperature-dependent assembly, and a mass-action regularization to suppress unbounded polymer growth. Validation across three representative cases—a simple dimer, a temperature-controlled octahedral shell, and a non-self-limiting polymerizing system—shows close agreement between optimized yields and molecular dynamics simulations, demonstrating robustness and transfer to finite-size canonical ensembles. The work provides a general design tool for programmable matter that integrates entropy, anisotropy, and concentration dependence, with potential applications to programmable colloids and magnetic handshake materials.

Abstract

Designing heterogeneous, self-assembling systems is a central challenge in soft matter and biology. We present a framework that uses gradient-based optimization to invert an analytical yield calculation, tuning systems toward target equilibrium yields. We design systems ranging from simple dimers to temperature-controlled shells to polymerizing systems, achieving precise control of self- and non-self-limiting assemblies. By operating directly on closed-form calculations, our framework bypasses trajectory-based instabilities and enables efficient optimization in otherwise challenging regimes.

Figures (7)

• Figure 1: Overview of our framework for designing heterogeneous, finite-sized assemblies.(A) Schematic of the full optimization and validation pipeline. The specification of candidate structures (e.g. stoichiometry, ground states) and initial parameter values $\theta_{\mathrm{init}}$ are provided to a closed-form calculation that predicts the equilibrium yield of all structures. The outputs of this calculation are directly differentiated to update the parameters via gradient descent, optimizing a user-defined objective function over these yields. The optimized parameters, $\theta_{\mathrm{opt}}$, are then validated through molecular dynamics simulations. (B) Internal structure of the differentiable optimization model shown above. We first apply a two-step analytical calculation to compute the equilibrium yields: (i) the partition function of each structure is computed, and (ii) these partition functions are mapped to equilibrium concentrations by numerically solving a self-consistent system of equations. The resulting yields are used to evaluate the objective function, $\mathcal{L}$. This entire procedure is implemented in a differentiable form, enabling automatic differentiation and gradient-based updates of $\theta$.
• Figure 2: Optimization and validation of a simple dimer system.(A) Schematic of the toy dimer system composed of two enantiomeric monomers, each containing three distinct patches. Only identical patch types interact, allowing the monomers to bind into a single target dimer configuration. (B) Convergence plot of the optimization toward the target equilibrium yield of $0.5$. The purple curve corresponds to initialization with a strong attractive potential ($\epsilon$ large), while the blue curve corresponds to a weak attractive potential ($\epsilon$ small). Insets show representative molecular dynamics snapshots corresponding to the system at those parameters. (C) Comparison between theoretically predicted yields and molecular dynamics simulations across many independent optimizations under varying conditions. In the left panel, interaction strength $\epsilon$ is fixed while temperature is optimized. In the right panel, temperature is fixed while $\epsilon$ is optimized. For most optimizations, the theoretical yield of the dimer at the end of the optimization was within $1 \%$ of the specified target value.
• Figure 3: Temperature-dependent control of shell assembly.(A) Illustration of the system of patchy monomers with two patch types (green and yellow) that self-assemble into a closed-shell structure. The optimization goal is to identify interaction parameters that produce switch-like behavior --favoring shell assembly at $kT_{\text{low}}$ and disassembly at $kT_{\text{high}}$. The red arrow indicates increasing temperature, highlighting the challenge of maximizing yield contrast across this range. (B) Set of possible assembly outcomes included in the calculation: the fully assembled target shell, a variety of incomplete off-target structures, and free monomers. (C) Optimization results. Left: Absolute difference in simulated fully assembled shell yields between $kT_{\text{low}}$ and $kT_{\text{high}}$ as a function of the percentage increase in temperature, computed using the optimized parameters. Right: Simulated yields of fully disassembled monomers at $kT_{\text{high}}$ for two optimization strategies. The green curve shows parameters optimized only to maximize yield at $kT_{\text{low}}$, while the red curve shows parameters optimized simultaneously to maximize yield at $kT_{\text{low}}$ and minimize yield at $kT_{\text{high}}$. Together, these results demonstrate the necessity of multi-ensemble optimization for achieving temperature-controlled assembly and disassembly.
• Figure 4: Controlling polymer growth through mass action regularization.(A) Schematic of the polymerizing system. On the left, we depict the the target $\alpha\beta\gamma$ trimer with the two strong interactions that define the desired assembly highlighted. The left panel depicts the $6 \times 6$ interaction matrix $\varepsilon_{ij}$ describing all pairwise interactions between the six patch types across the three monomer species ($\alpha$, $\beta$, and $\gamma$). (B) Illustration of the challenge posed by non-self-limiting polymerization. All monomers, dimers, and trimers can be explicitly included in the optimization (left). However, the space of possible longer chains ($n \geq 4$) grows combinatorially and cannot be exhaustively enumerated (right). This makes it intractable to directly account for every off-target structure in the analytical calculation. (C) Simulated equilibrium yield distributions by cluster size as a function of inverse temperature ($1 / kT$). Simulations were performed with optimized parameters obtained both without (left) and with (right) mass action regularization. Callouts depict representative simulation snapshots at $kT = 0.75$.
• Figure S1: Convergence behavior of the optimization with desired yield of 1.0. The dimer yield fully converges to near 1, value around the 600th iteration. The red curve shows the average attractions strength parameters that are being optimized. after a steep increase curve to $\epsilon_{avg} =4$ the value keeps increasing slowly above 4.5 even after yield convergence has been reached.