High-Dimensional Computation in Soft Matter

We present a machine-learning approach to predict the interactions among IDPs in heterogeneous mixtures from sequences. Our approach predicts partition coefficients and multicomponent phase diagrams with quantitative accuracy by going beyond the pairwise approximation and ad hoc selection of sequence features that underlie existing methodologies. By applying our approach to state-of-the-art IDP force fields, we demonstrate that the highly simplified representation of IDP sequences that is learned by our model completely determines the thermodynamic behavior of mixtures with arbitrary numbers of IDP components and compositions. Moreover, we show that Euclidean distances in the learned representation are directly proportional to the differences in the chemical potentials of IDP sequences in arbitrary mixtures. This property of our model establishes a physically meaningful method to quantify the interaction specificity of IDP sequences, to predict the effects of sequence mutations, and to perform coevolutionary analyses. Our approach therefore provides a generalizable, interpretable, and quantitatively accurate characterization of IDP interactions that can be straightforwardly applied to both simulation models and high-throughput experiments.

Liquid mixtures can separate into phases with distinct composition. This phenomenon has recently come back to prominence due to its role in complex biological liquids, such as the cytoplasm, which contain thousands of components. For simple two-component mixtures phase-separated states are global free energy minima. However, local free energy minima, i.e., metastable states, are known to play a dominant role in complex systems with many components. For example, Hopfield neural networks can retrieve stored information from partial cues via relaxation to metastable states. Under what conditions can phase separated states be metastable, and what are the implications for information retrieval in multicomponent liquids? In this work we develop the general thermodynamic formalism of metastable phase separation. We then apply this formalism to an illustrative toy example inspired by recent experiments, binary mixtures with high-order interactions. Finally, as core application of the formalism, we study metastability in Hopfield liquids, a class of multicomponent mixtures capable of storing information on the composition of phases. We show that these phases can be nucleated from partial cues via metastable phase separation. Spatial simulations of liquids with a large number of components match our analytical solution. Our work suggests that complex biological mixtures can retrieve information through metastable phase separation.

Biological systems often contain matter that can be found in multiple different configurations and structures, depending on internal and external conditions. Systems, like DNA-tile systems, that can self-organize into multiple target structures from the same set of constituent particles are called multifarious and can be used for computation [1]. For sequential information processing, however, it is necessary to actively switch between different configurations and doing this in a controlled and dynamical manner requires out-of-equilibrium physics in the form of non-reciprocal interactions [2]. We perform Monte Carlo simulations on DNA tiles with non-reciprocal interactions to induce transitions between multifarious target structures and show that these can perform basic computations, akin to the operation of finite-state automata. We demonstrate that such systems can be designed to perform a variety of tasks including counting, computing the modulo of an input number, and recognizing input patterns. This approach naturally integrates memory, sensing, and actuation within a single physical platform, offering a route toward low-energy physical computation in microscale systems. [1] C.G. Evans, J. O'Brien, E. Winfree & A. Murugan, Nature 625 (2024) 500-507 [2] S. Osat & R. Golestanian, Nat. Nanotech. 18 (2023) 79-85

We firs outline the SAT-assembly method: a computational pipeline that solves the inverse design problem: Given a target material or finite-size structure, find a set of building blocks that self-assemble into the target shape while avoiding kinetic traps and other competing free-energy minima. The method relies on formulating the design problem as a Boolean Satisfiability Task, for which there are efficient algorithms available, and using multiscale modeling to simulate the structure self-assembly. We apply the method to design of several highly coveted colloidal self-assembled materials, including pyrochlore lattice, and hexagonal diamond and diamond cubic lattices, which have promising applications in photonics. We also show how this method can be applied to finite-size structures, and introduce a universal polycube platform, where arbtirary polycubic shape can be otained by self-assembly from "minimal kit", the smallest number of objects that guarantee assembly in high yield. We will explore the physics of self-assembly of multicomponent systems, including an example of two-component self-assembled material which acts as a counterexample to classical nucleation theory. To show the applicability of our modeling result, we will also show that the computationally designed materials can be realized from self-assembled DNA building blocks. We verify successful experimental realizations of crystals and finite-size objects using TEM and SEM imaging, as well as with SAXS and cryoEM.

A major hypothesis in systems biology has been that cellular computation occurs through discrete modules and characterizing these modules might unlock powerful theories of cellular function. This picture looks increasingly incomplete in eukaryotes. Many interactions of interest are high-dimensional, mediated by diffuse coupling among many types of molecules, and circuits are broadly linked through shared enzymatic machinery. This raises the question of what computational principles underlie such implementations, and whether we can ever build effective models for studying them. In this talk I will describe recent research along two fronts, long-term biological memory and cellular differentiation. For the first, I will present evidence that long-term memory in immune and epigenetic contexts occurs through critical tuning of "memory units", such as silenced genomic regions and immune memory cells, and show how competition between these units allows these systems to selectively retain functional components. For the second, I will present a model of cell identity control that explains how cells encode identities as attractor states, and that quantitatively accounts for the origin of complex differentiation landscapes across different systems. In both cases I will highlight how the interplay between specific and non-specific interactions underlies the emergence of complex functional behaviour.

How can soft and living matter systems perform statistical inference under constraints of noise, delay, and limited memory? We show that optimal adaptation in time-varying environments does not require forecasting future states, but instead emerges from tracking an actionable target defined by the instantaneous optimum and its rate of change [1]. This reformulation separates inference from prediction and reveals fundamental limits imposed by delayed information. Using a dynamics-informed neural network framework [2], we demonstrate how biological systems can approximate the actionable target from past observations. We further show that recurrent structures, such as circadian and circannual clocks, dramatically reduce memory requirements by encoding environmental regularities, enabling near-optimal performance with minimal data. These results provide a physically grounded perspective on inference in living matter and suggest concrete routes to implement adaptive algorithms in soft-matter systems and biomolecular circuits. [1] J.M.G. Vilar and L. Saiz, Actionable forecasting as a determinant of biological adaptation, Advanced Science, 12, 2413153 (2025). [2] J.M.G. Vilar and L. Saiz, Dynamics-informed deconvolutional neural networks for super-resolution identification of regime changes in epidemiological time series, Science Advances, 9: eadf0673 (2023).

Adaptive behavior in living systems relies on the interplay between information processing and reproduction: information processing enables learning, and reproduction gives learning its value through reproductive success. How such capabilities can arise and become coupled in systems built from simple physical ingredients presents an open engineering and evolutionary challenge. I will describe a minimal reaction–diffusion model that exhibits specific adaptation to complex temporal sequences of environmental changes. In this system, learning and reproduction emerge jointly from simple non-equilibrium chemical dynamics, without prior design or evolution.

I will give an overview of our recent work on the effects of mobilities in phase separation kinetics, including (i) approximating mobilities of multi-species system from single particle and collective structure factor data, (ii) avoided exceptional points and first order transitions in the spinodal phase diagrams of non-reciprocal systems, (iii) interface-localised spinodal dynamics in systems with glass-like mobility constraints.

Liquid-liquid phase separation provides a significant mechanism for spatial organization in living and synthetic systems. Diverse interactions between many constituent components can generate complex hierarchical structures, including DNA nanostar droplets with programmable interactions and condensates in the nucleus with multiple subcompartments. While phase separation has been extensively studied for two fluid phases, the principles governing the evolution of many component mixtures still remain unclear. In this work, we show that the phase separation dynamics of multicomponent fluid mixtures are fundamentally connected to mathematical coloring theorems, because interfaces between identical adjacent fluids are quickly eliminated through coalescence. Using numerical simulations, we find that in confined domains, coalescence driven hydrodynamics is suppressed when four or more phases are present, and subsequently derive a theoretical model for the phase separation dynamics from graph coloring theory. In unconfined three dimensions, by contrast, suppression of coalescence driven hydrodynamics emerges only asymptotically with increasing numbers of phases, reflecting a change in coloring rules for non-planar arrangements. By modifying the interfacial tensions, we can restrict the allowed fluid adjacencies and produce highly complex coarsening dynamics that differ for each phase.

Biomolecular condensates organize the cellular interior without enclosing membranes, yet many appear at precise positions where they perform specific biological functions. In this talk, I will discuss how such positioning can be understood through local thermodynamic control. I will focus first on the canonical biomolecular condensates, P granules in the one-cell C. elegans embryo, where these condensates segregate to the posterior by position-dependent dissolution and condensation. I will show how a biochemical gradient can be interpreted as a spatial modulation of the local saturation concentration, and how a temperature gradient can substitute for this endogenous cue to rescue P-granule segregation. This reveals a form of thermodynamic interoperability, in which distinct fields produce equivalent positioning outcomes by acting on the same local phase boundary. I will then give another example from tight-junction assembly, where cells position ZO-1 condensates through local surface condensation to form a functional epithelial barrier. As an outlook, I will discuss ongoing work on P-granule wetting of the nucleus. Together, these examples suggest that cells can manipulate local thermodynamic parameters to organize condensates in space.

In the 1980s, the field of AI was focused on design—the goal was to achieve complex collective behavior by programming it. This paradigm was displaced by the current one, in which networks with many tunable degrees of freedom are optimized to learn by example. I will argue that it is time to shift the paradigm in physical systems, including biological ones, by thinking of complex function as emerging not from design but from optimization of systems made of tunable matter: collections of physical components that interact with each other in ways that can be adjusted to produce collective behavior. I will extend current scalable tuning processes from ones that apply strictly to systems with reciprocal interactions to systems with non-reciprocal interactions, allowing the systems to perform complex dynamical tasks.

Biological systems exhibit a unique ability to design diverse structures from a shared set of building blocks, with a plethora of proteins made from a limited set of amino acids as a prime example. Furthermore, these systems often use building blocks efficiently by introducing transformations between different structures. A self-assembled structure might undergo structural transformations, disassemble, and form a new structure with different functional purposes, without the need to discard the current structure and start anew. To unravel this mystery, one must examine the underlying non-equilibrium processes that make this shape-shifting behavior feasible. Here, we leverage non-reciprocal interactions between building blocks to provide a foundation for designing dynamic shape-shifting structures. We used a multifarious self-assembly model, which is the molecular counterpart of the celebrated Hopfield associative memory. This model treats desired target structures as memories that should settle in the local minima of the energy landscape. The model achieves this through a set of learning rules that output the interaction matrix between building blocks. Triggering the system with a small seed leads to the proper self-assembly of the structure on demand, making the system a molecular associative memory. By upgrading the multifarious self-assembly model to its non-equilibrium counterpart with non-reciprocal interactions, we introduce the ability to not only self-assemble different structures on demand but also facilitate shifts and transformations that lead to shape-shifting behavior. Non-reciprocal interactions between building blocks break action-reaction symmetry, providing the necessary non-equilibrium conditions that drive shape-shifting behavior.

Many essential cellular machines, such as ribosomes and spliceosomes, consist of dozens of distinct proteins that must rapidly find their correct partners amid numerous alternatives. Yet, these protein complexes assemble both accurately and fast. From the perspective of equilibrium self-assembly driven by binding energies, high speed and high accuracy are incompatible. In fact, extensive biological evidence points to the central role of assembly factors that transiently catalyze specific binding events. Here, we develop a minimal non-equilibrium self-assembly model that captures how variations in binding rates shape assembly pathways. Within this framework, large multi-component structures are encoded as kinetic traps that can be transiently retrieved from small assembly seeds. Such kinetic encoding overcomes the intuitive speed-accuracy tradeoff, allowing fast and accurate growth. The model reveals that slow and error-prone binding events create speed and encoding bottlenecks that can be overcome by targeted catalysis. Together, these results offer a unified explanation for three hallmark features of the assembly of large heteromeric protein complexes: high speed, restricted pathways, and widespread use of assembly factors.

Experimental platforms for assembling nanostructures out of programmable building blocks are developing at an increasingly rapid pace. Nevertheless, our ability to create complex, precise, and functional nanostructures pales in comparison to the “high-tech nanotechnology” found throughout biology. To make progress, we need to understand how a moderate number of component building blocks can be programmed to assemble into large and precise structures, where each component appears multiple times. However, this economical approach often requires binding to be stochastic, meaning that the bonds that form are not deterministically set by the particles’ design. Despite the considerable complications this creates, I will show that this is nevertheless a powerful and scalable regime for complex assembly. Furthermore, one can predict exactly what structures can be assembled at high yield – and how to achieve them – through a high-dimensional polyhedron imposed by thermodynamic constraints. I will also discuss the dynamics of assembly, and reveal how we can understand and hopefully avoid long-lived kinetic traps.

The field of active matter has established that internal activity and broken reciprocity can produce collective motion, unconventional mechanical response, and emergent organization. A central question is whether these same ingredients can enable computation and inference in active matter systems. To address this, we have developed a minimal experimental platform of active robotic chiral spinners, inspired by odd living crystals, with long-range medium-mediated coupling and tunable nonreciprocal interactions. We treat spinning frequency, which sets the strength of nonreciprocity, as a learning degree of freedom that modifies effective interactions. Using reinforcement learning in a centralized training, decentralized execution framework, we show that agents using only local observations can achieve coordinated tasks such as navigation. To interpret these behaviors, we apply mode decomposition methods that identify low-dimensional structures underlying learned collective strategies. In parallel, we are developing physical learning rules in which these control parameters evolve through local feedback from medium-borne signals, allowing interactions to become dynamical variables. This work aims to establish a framework for emergent computation in active matter, where distributed, nonreciprocal interactions enable materials to process information and generate function.

How can a physical system find a desired shape among a large number of possibilities? We address this question through the lens of folding, viewing shape formation as a search and navigation problem in a high-dimensional configuration space. I will discuss recent work spanning microscopic self-folding kirigami to mechanically driven macroscopic structures. In self-folding systems, different two-dimensional nets leading to the same target shape can exhibit markedly different folding yields and times, revealing the importance of topology and competing folding pathways. To understand these effects, we developed a general algorithm that systematically maps folding configuration spaces and the transitions connecting them. The resulting landscapes provide a global description of the accessible folded states and suggest strategies for controlling shape formation. As an example, I will show how external vibrations can steer centimeter-scale nets toward different folded configurations, with the selected structure depending on the amplitude and frequency of the forcing. These results establish a direct connection between geometry, dynamics and control in folding systems. More broadly, they suggest that folding can serve as a prototype model for studying how driven nonequilibrium systems explore, process information, and navigate complex high-dimensional landscapes.

Biological organisms adapt to changing environments by evolving diverse strategies to extract and use information. These strategies can be viewed as computational algorithms that process environmental information to infer adaptive phenotypic responses. In this talk, I will present examples of such algorithms with different forms of memory and computation. In unpredictable environments, a population can learn the long-term statistics of the environment through a transgenerational feedback, which acts as gradient descent and allows the population to optimize its phenotypic composition. When environmental variation has a temporal structure, organisms can utilize their internal states as an effective memory, which operates like a finite-state machine to predict future environments. In situations where organisms use a noisy cue to infer environmental transitions, their optimal response forms a low-dimensional environment-to-phenotype mapping, which can be described by a neural network that supports different adaptation strategies. These examples illustrate how living systems can implement adaptive algorithms that provide evolutionary advantages in changing environments.

Biological collectives such as schooling fish and insect swarms perform robust, scalable decision-making and probabilistic estimation through effective physical or social interactions and motile states. We explore how simple active matter systems can be engineered to implement distributed statistical inference without centralized control. We map physical degrees of freedom to statistical primitives, derive minimal interaction rules that realize collective behavior, such as consensus decision-making, observed experimentally in noisy environments and under competing cues. Our outcomes aim to unify principles of biological collective cognition with soft condensed-matter design, yielding new paradigms for programmable materials, adaptive sensing, and distributed robotic systems that perform inference via embodied physical dynamics.

Complex systems, such as disordered materials, are composed of many interacting microscopic degrees of freedom, and modifying these degrees of freedom intelligently can enable new complex functionalities. A central challenge is to modify the microstructure in a scalable manner, specifically, the limit of large systems. Recent ideas inspired by learning theory offer a distributed route: local evolution laws can allow a material to autonomously adapt its internal structure so as to achieve a target function. In this sense, a material learns to solve its own design problem. While most previous work has focused on quasistatic settings, the dynamical regime offers a broader range of functionalities that depend on the sequence of applied inputs, exhibit rate-dependent behavior, and realize responses that are inaccessible in quasistatics. In this talk, I will discuss how physical learning ideas can be extended to the dynamical regime, governed by damped Newtonian dynamics. I will discuss key considerations for controlling dynamics and introduce several learning rules. Finally, I will show how simple mechanical and electronic systems can acquire complex functionalities, including classifying temporal signals, recognizing spoken words, and tuning the rate-dependent response of materials.

Programming the self-assembly of colloids into non-crystalline materials with prescribed organization remains a major challenge. Conventional passive, irreversible self-assembly usually evolves slowly or incompletely to the desired state because it explores a vast multidimensional landscape with numerous kinetic traps. Here, we circumvent the curse of dimensionality and kinetic arrest by encoding specific, reversible interactions between DNA-coated Brownian emulsion droplets that can freely rearrange after binding [1]. The type and strength of DNA-mediated interactions are programmed via enzyme-driven, out-of-equilibrium reaction networks [2], which provide precise spatiotemporal control over assembly pathways. This system allows us to encode the formation of self-limited clusters with diverse shapes and composition, such as ribbons, flowers, or checkerboards (Fig. 1b), and to drive morphological transitions that are inaccessible to fractal aggregates formed from solid colloids [3]. For instance, we can induce the transformation of floppy clusters with checkerboard motifs into rigid clusters with core-shell structures (Fig. 1c), reminiscent of differential adhesion-driven cell segregation in biological tissues [4]. Altogether, this droplet-enzyme hybrid system provides a powerful route to engineering dynamic, reconfigurable colloidal architectures.

Cells operate by coordinating reactions to occur at the right time and in the right place in a biochemically and mechanically complex environment. One remarkable way cells organize biomolecules to perform reactions is by utilizing the physics of binodal coexistence, called 'liquid-liquid phase separation' (LLPS). Given the complexity of living systems, we develop controlled model systems to investigate their dynamics and structures. LLPS model systems built from DNA nanostars can harness the design power of nucleic acids to display a range of near-equilibrium dynamical and structural behaviors, including spatial patterning, nucleation dynamics, size control, and switchable gel mechanics. Finally, inspired by transcriptional condensates in the cell nucleus, we develop a non-equilibrium model system in which nanostar condensation controls and can be controlled by in vitro transcription reactions. Precise models of LLPS systems, composed of enzymatically active droplets, will not only illuminate general organizational principles in living systems but also enable the development of new modes of biomolecular computation and mechanical transport.

Metabolism is a fundamental module of life whose physical and chemical constraints make it amenable to quantitative measurement and mathematical modeling. A standard theoretical approach is flux balance analysis (FBA), which hypothesizes that metabolic networks maximize an objective function such as biomass production. FBA scales well computationally but largely ignores the complexity of single-cell heterogeneity and interactions. I will discuss theoretical and experimental approaches to address these limitations, drawing on ideas from statistical mechanics. Starting from simple E. coli experiments, I show that a maximum entropy inference framework à la Jaynes outperforms FBA in predicting fluxes, recovers single-cell growth-rate distributions, and yields theoretical insight into underlying trade-offs (Nature Communications, 2018). I then turn to single-cell experiments in tumor–stroma co-cultures exhibiting the Warburg effect — aerobic acidification — probed with pH-sensing nanofibers (ACS nano, 2022). These reveal that acidification arises from leakage in a lactate exchange network; a maximum-entropy constraint-based model that allows for intercellular interactions shows that the overflow threshold is a bona fide phase transition separating balanced and leaked regimes (Communications Physics, 2025; Science Advances, 2025). Next, I show that an analogous interacting model consistently interprets nanoscale mass-spectrometry data from the cyanobacterium Synechocystis, suggesting cells self-organize into two sub-populations — autotrophs and mixotrophs — where the latter are fed by the former and carry out replication (arXiv, 2025). Finally, returning to E. coli, I extend the maximum entropy framework to dynamical regimes and show that recently observed co-consumption during glucose-acetate diauxie occurs at the single-cell level rather than through sub-population exchange (bioRxiv, 2026). Unlike FBA, which reduces to linear programming, the modeling and inference problems addressed here are computationally harder — analogous to statistical mechanics of disordered systems, particularly Gardner problems — and require more sophisticated methods such as optimized Monte Carlo, cavity methods, and mean-field approximations.

Machine learning based on computation performed through the dynamics of simple physical systems, rather than manipulation of transistors, is gaining popularity, with the hope of building machine learning systems orders of magnitude more energy efficient than silicon based computers. The physical nature puts severe limitations on the range of weights and the activation functions of layers (if they are present in the first place). However, many of them do not perform better than simple single layer perceptrons. In this talk we explain why this is the case and propose new ways forwards by laying out the minimal requirements for physical machine learning.

Biological cells use phase separation to organize their interior and regulate a wide range of functions. In this talk, I will discuss how the physics of multicomponent phase separation can help us understand both intracellular organization and information processing. In the first part, I will address the challenge of determining multicomponent phase diagrams. Ι will present recent numerical approaches and discuss scaling behavior of multicomponent mixtures. In the second part, I will explore the idea that cells may exploit phase transitions to process information, using condensates not only as compartments but also as dynamic computational elements. I will conclude with an outlook on open directions, including the roles of energy-consuming processes.

Learning to couple responses to signals is key for autonomous adaptation to changing environments. In biology, this is often achieved through Pavlovian conditioning, where a neutral conditioned stimulus is trained to elicit the same response as the unconditioned stimulus. Such mechanisms are largely absent in physical systems, yet they represent a crucial first step towards autonomous learning materials. Here, we combine a programmable colloidal system with molecular programming to encode specific responses, analogous to Pavlovian conditioning. Our system stores and exploits associative memory to select responses to conditioned signals, marking a step towards materials capable of in-situ, experience-driven adaptation.

Folding occurs along pathways in a rugged, highly interconnected free-energy landscape, where structures are strongly coupled to each other through their backbone conformations. This coupling hinders the targeting of specific structures, yet simultaneously enables interconversion between conformations. To control folding thus requires identifying and exploiting viable folding pathways. Here, we study folding pathways in a minimal model system of colloidal droplet chains with programmable DNA-mediated interactions: colloidal foldamers. Through simulations, theory, and reinforcement learning, we uncover folding pathways leading to targeted geometries. Surprisingly, these pathways are not strictly downhill; instead, they can pass through intermediate misfolded states that subsequently refold into the target geometry. We translate these pathways into non-monotonic experimental protocols for ensemble self-assembly, paving the way toward systematic folding-based synthesis of functional metamaterials.

We investigate how cyclic environmental variation can induce learning and memory in adaptive soft-matter systems. Using inverse-designed disordered solids trained to achieve target mechanical properties over a range of values, we find that the system evolves toward a marginally absorbing manifold, where behavior is reversible within the training range but irreversible outside it. This transition encodes a memory of past environments. We propose a general mechanism for memory formation based on discontinuities in the gradient of the trained response, providing a general framework for learning and memory in adaptive physical systems.

Despite the unsustainable growth in energy consumption by artificial intelligence models and the recognition of the major role played by metabolic constraints in brain evolution, the relationship between computation and energy remains insufficiently studied and understood. Recently, Padamsey et al. experimentally investigated this relationship in the context of visual information processing in food-deprived mice. Combining analysis of their activity data and modeling inspired by statistical physics, in particular some variants of the Hopfield model, I will propose some mechanism by which neural circuits can spare considerable energy with little impact on their performance.