To the editors:

Individuals, our society, the economy, linguistics, ecology, and many other things operate in a fascinating, perplexing, and mysterious realm far from thermodynamic equilibrium. The systems in this realm can modify and be modified by their surroundings. These are dynamic adaptive systems—like living organisms, the stock market, weather, and crowds. None are fully understood: even experts do not quite know why these entities work the way they work, which direction they will evolve, or how to steer them toward a desired goal. When asked, a layperson intuitively answers that this is so because they have many interacting parts—whether atoms, cells, people, stock exchangers, etc.—with intertwined relationships, and there is always uncertainty about their choices among multiple options.

As a scientist, I am amazed to see how people of different ages, with differing academic degrees, and from different professions can accurately describe far-from-equilibrium, or dissipative, systems. Each group uses quite different words. In physics, these would be described as many-body systems with local and nonlocal interactions that constantly modify and are modified by stochastic events and strong nonlinearities. One missing piece in the layperson’s description is that these systems are irreversible, perhaps because this is self-evident. If one watches a movie of their evolution played forward or backward in time, the difference is obvious, unlike the motion of planets or the swinging of a pendulum. Humans do not evolve back to a cell, a tree does not grow back to its seed, nor does technology disassemble back into its constituent parts. But why not? Although we can give incomplete qualitative and intuitive answers to this question, we are not yet able to provide quantitative and factual answers. This conundrum goes back to the early days of statistical physics. Our mathematical and physical toolbox is best suited to treat thermodynamic equilibrium or near thermodynamic equilibrium conditions. In this domain, the nonlinearities—mutually affected choices, patterns, directions, structure, behavior, etc.—are few, stochastic events—fluctuations, randomness, noise, errors, deviations, etc.—are weak, and everything stays the same on average as time goes by.

Several decades ago, researchers such as Lars Onsager, Ilya Prigogine, Hermann Haken, and Alan Turing widened our perspective on how flows and forces are related under nonequilibrium conditions and how patterns are formed in natural and driven systems.¹ The 1990s witnessed major theoretical developments that kickstarted the field of stochastic thermodynamics, notably the formulation of the Jarzynski equality and Crooks fluctuation theorem, which showed how to form links between the work done under nonequilibrium conditions and equilibrium quantities such as free energy.² These two outstanding results were quantified and demonstrated empirically.

Recently, these fluctuation theorems have been extended to explain why time-irreversible trajectories of evolution are more probable from the thermodynamics point of view and the dissipative adaptation idea of Jeremy England.³ Surely, results will soon be available from empirical studies that put these notions to the test.

Despite these encouraging developments, deep questions still linger: When far from equilibrium, in the presence of fluctuations and faced with multiple steady states with small energy differences, how does a system evolve? And why? To find the answers, we need experiments performed on physical systems, first to test the emerging theories, and second to help us develop brand new ones.

In recent years, researchers have become increasingly vocal about the importance of empirical evidence to map the uncharted territory far from equilibrium. Among others, our team sprang up to meet this challenge and performed several experiments along these lines. One of our discoveries was the universality of dissipative self-assembly. We devised a self-assembly method that works for any material suspended in any liquid, such as few-nanometers-large quantum dots in water,⁴ nanometer-sized polystyrene spheres in water, and micrometer-sized living organisms in their growth media. We found that regardless of whether the material is living or nonliving, heavy or light, large or small, complex or simple, the emergence and growth of its aggregates far from equilibrium obey the same physical rules.⁵

The Universality of Dissipative Self-Assembly

Our study has attracted significant attention and was critiqued by Michael Engel in an Inference article published in the journal’s May issue.⁶ After providing a well-articulated background to dissipative self-assembly, he expressed concerns about our analysis methods to show the universality of dissipative self-assembly—namely, scale-invariant autocatalytic sigmoidal growth and the Tracy–Widom (TW) statistics. Engel then shared his perspective on how neural networks may inspire the development of similar techniques that can train chemical reaction networks for controlled self-assembly of desired structures and functionalities. Several months later, a response by Tanja Schilling was published in the journal’s September issue.⁷ She brought up her disagreement with Engel’s statement that “there is no general extremum principle that governs the evolution of systems far from equilibrium.” She also posed an interesting question: Is the timescale separation between different degrees of freedom at the origin of the universal behavior that we documented? Our response will address Engel’s two criticisms, answer Schilling’s question, and add our perspective on the best way to establish the cartography of the far-from-equilibrium landscape.

Scale-Invariant Autocatalytic Growth

In his review, Engel writes, “Filling area is not a convenient measure to study growth because it does not increase linearly, even in static self-assembly.” Putting aside the fact that Engel does not offer any alternative measures to study aggregate growth in a finite experimental system, it appears that he has missed a crucial point: our analyses were explicitly designed to test our own analytical model that describes how self-sustaining autocatalytic aggregates could form and grow in a strongly stochastic and highly nonlinear system. Based on coupled Brinkman–Forchheimer and Fokker–Planck equations, the model explains how particles, dragged by strong flow fields, can collect at a physical boundary without scattering from it. Considering that their constituents are strongly Brownian, why do these collections grow rather than disaggregate? The model provides an answer by describing the intrinsic feedback loops between the flow fields, aggregate, and Brownian motion.⁸

According to our model, the aggregate is unstable and will dissolve when its size falls below a certain threshold. This part forms the bottom, flat-line of the S-curve. Once this threshold is reached, the aggregate starts to grow, forming the middle part of the S-curve. While growing, the aggregate reduces the velocity of the flow fields because the voids left between the particles filter the fluid, acting as a sieve.⁹ The sieve effect remains localized and extends a few micrometers beyond the aggregate periphery, where the fluid flows are slowed down sufficiently. When a dragged particle enters this region, it also slows down and adjoins the aggregate rather than scattering from it. Outside this region, the flows remain strong, continuously supplying new particles to the aggregate. A larger aggregate extends the surrounding slow-flow region, which increases the probability of a larger number of particles entering the region and adding to the aggregate. The finite extent of the laser-induced flows, a limited number of particles, and Brownian motion necessarily terminate this constant growth at a finite size, eventually leading to a saturation point that forms the upper flat-line of the S-curve.

To test the validity of this model, which describes the filling ratio, we have chosen a region of interest (ROI) and calculated its filling ratio with the aggregate over time. We have experimented with different aggregate sizes formed by tens to thousands of particles, chosen whole aggregates as the ROI, or defined an ROI within the aggregate with different sizes, shapes, and positions to test the limits of our model. All variations follow a sigmoidal curve. Some of these variations are presented in our Nature Physics paper.

We are not surprised by the sigmoidal growth of our aggregates, as it is also manifested in numerous other systems far from equilibrium, even in the social sciences, economics, epidemics, and tumor or nanocrystal growth. The common point in each of these examples is the presence of action-competing intrinsic feedback loops, just like in our case. Similar growth behavior in a system with externally induced feedback loops is unlikely to occur. Such a result would require complete control over astronomically many degrees of freedom designed to drive the system dynamics. Inherent feedback loops save all the hassle by managing this process internally; they mutually lock the vast majority to each other through their nonlinear interactions. One can control this machinery once the dynamics forming the loops are identified correctly.

The sigmoidal autocatalytic growth kinetics come about as a result of the inner dynamics of our system controlled by the intrinsic feedback loops. An excellent example and existence proof of the possibility of controlling a system with a vast number of degrees of freedom is the mode-locking of a laser, which involves nonlinearly locking the phases and amplitudes of thousands of electromagnetic modes in an optical cavity. Mode-locking is not only a fascinating example of self-organization, but is also a mature and robust technology underlying thousands of experiments every day.

Tracy–Widom Statistics

A second point Engel raises in his review concerns our fluctuation analysis:

The extraction of TW statistics is reliant on high-pass filtering in the form of a temporal span analysis. The agreement of the data up to the eighth moment of the distribution is certainly a stringent test. Even so, the analysis undertaken required choices of the temporal span parameter in a certain window and was performed for only a single particle type.

It should be noted here that we aim to analyze the interface fluctuations of the growing aggregates. The characteristic timescale of the fluctuations is relatively short compared to that of the growth, and because the fluctuations are riding on the growth curve, one has to separate the two timescales using a temporal span parameter or a cut-off frequency. This is a well-known procedure in time-series analysis. It is the only logical way to correctly separate the steady growth component from the fluctuations, which is, at the same time, practically applicable to thousands of experimental image frames. Contrary to Engel’s assertion, the qualitative results do not change when the temporal span parameter is chosen differently. In our study, we present how the statistical moments vary within an already extensive range of temporal span values, ~200 to ~1,800ms.

Like the sigmoidal growth curve, finding TW statistics in our experiments was not surprising. In our earlier Nature Communications paper, we showed that the probability distribution of the fluctuations was in accordance with Gaussian statistics, as expected before we drive the system with an ultrafast laser.¹⁰ Once the laser was driving the system away from equilibrium, we observed the emergence of giant number fluctuations. We already knew that our particles were correlated once the laser-induced flows collected them, so the fluctuations should not be in accordance with Gaussian statistics. We then searched for possible non-Gaussian statistical candidates applied to the systems with correlated constituents. TW statistics were a perfect match, which is increasingly commonly observed in physically diverse systems. As also noted by Engel, we strenuously tested the mathematical validity of our findings by establishing an agreement up to the eighth significant moment. Researchers involved in all prior empirical demonstrations of TW statistics were content with only the first two, but we did not stop there. We tested the necessity of inter-particle correlations to obtain TW statistics and showed that if the particles remain uncorrelated, Gaussian statistics are recovered. In short, no correlations, no TW statistics.

Engel criticizes our TW analysis for providing the statistics for a single particle, polystyrene spheres. In our paper, we explain this point, but we can discuss it here in more detail. It is possible to perform similar analyses on living organisms, but this would require prohibitively long experiments to collect enough statistics—not to mention the significant time and effort necessary to culture and grow the cells. More importantly, organisms are non-uniform and have soft shells with ever-changing sizes and topologies. As a result, it is difficult to optimize the detection algorithms and computational analysis tools for each experimental image frame—typically more than 50,000 frames per experiment—to correctly separate the growth and fluctuations components of the cell aggregates. Such work would fill another doctoral thesis, if not two. As for the quantum dots, here we are faced with a fundamental limitation. The Abbe diffraction limit prevents us from resolving the particles and extracting the fluctuation information. For the moment, at least, this is out of the question.

Yet none of this is truly necessary. The aggregation of all particles and organisms exhibits non-Gaussian statistics when driven by the laser, and individual cells and quantum dots are correlated within an aggregate, precisely as in the case of polystyrene spheres. Both the sigmoidal growth kinetics and the TW statistics emerge from the core dynamics of this dissipative self-assembly system and its inherent feedback loops. Indeed, it is important to note that one cannot deliberately arrange to obtain a sigmoidal growth curve or TW statistics: in our experiments, the only control we have over the system is the laser power and beam position. With merely two control knobs, one cannot dictate to thousands of particles where to go or what to do.

Unlike other studies, our polystyrene particles are not decorated with functional materials that respond to external fields. The liquid solution is simply water; the glass slides are not pre-treated with chemicals or any functional agent. In the case of organisms or quantum dots, the solution is also chemically different. Every experiment is different as the system is strongly stochastic, and one cannot repeat a sequence of events observed in a single experiment. Still, we saw and reported that none of these matters, and, despite all the differences, emergence and growth of the aggregates obey the same physical rules. Quoting Schilling,

[I]rrespective of the nature of the forces between the aggregating units and the number and nature of their internal degrees of freedom, the evolution of certain macroscopic observables—such as the size of aggregates and the magnitude of surface fluctuations—is governed by the same equation of motion.

Dimensionality

In his review, Engel also writes:

The experiments described in the paper are limited to particles constrained between two glass plates in a quasi-two-dimensional system. Further confinement is accomplished in some cases by means of a cavitation bubble that nucleates spontaneously due to localized boiling of the solution. Determining whether a similar degree of control of convective fluid flow is possible in a three-dimensional setup requires further experimentation.

Our results included quantum dots approximately 3 nanometers large. We do not yet have the technology to arrange the liquid film thickness to be comparable to a single quantum dot size. Therefore, for individual quantum dots, the space between the two glass slides is immense: ~1 micrometer, thick enough for hundreds of layers. And indeed, they are moving in 3D when not driven by the laser. It is clear from the experimental videos that when the system is driven by the laser, quantum dots are dragged by Marangoni flows toward their collections, similar to the polystyrene spheres and living organisms.

The critical issue here is not the system being 2D or 3D, but how the symmetry is broken by the injected energy. We break the symmetry only in the plane of the field view of the experiments and not along the third dimension. This symmetry breaking necessarily results in Marangoni flows even if the thickness of the bulk liquid is significantly increased. This is because the laser pulses are absorbed via the multiphoton absorption processes at the glass-liquid interface, and in fact mainly by the glass. This absorption and energy injection is highly localized in space and time, thanks to ultrafast pulses. Neither the bulk of the glass slides nor the liquid receives much energy at all. An empirical demonstration of this unique capability is depicted in the videos accompanying our paper, where we showed that whatever the beam does and wherever it goes, particles and organisms follow.

The Origin of Universal Behavior

As summarized by Schilling, commonly observed phenomena in diverse systems are understood and put into a mathematical framework through identifying their characteristic time, energy, and length scales. In her letter, Schilling asks whether the timescale separation between different degrees of freedom originates from the universal behavior that we documented. Using a thought experiment, comparing the equations of motion for quantum dots and M. luteus bacteria, she concludes that they are expected to differ. She then says, “There is something about the choice of self-assembly protocol in Makey et al. that removes the differences between these equations of motion.” Later she adds, “For theoreticians, the interesting question brought up by experiments such as those presented by Makey et al. is how to generalize this kind of result to systems far from equilibrium.”

While addressing Engel’s criticisms we have also provided answers to Schilling’s questions: the origin of the universal behavior we described is the intrinsic feedback loop that controls the inner dynamics of our system, following the slaving principle introduced by Haken.¹¹ As we also write in our paper, this process mutually locks astronomically many degrees of freedom through nonlinear feedback mechanisms.

We did not explicitly measure the characteristic time, energy, and length scales of all the processes playing a role in aggregate growth in our experiments because these quantities cannot be compared for the different materials that we experimented with. Liquid media have different chemical compositions and surface tensions that directly affect the velocity of the flow fields. As a result, we tune the laser power to different values for experiments performed with different materials. Unlike polystyrene particles, living organisms are active and tend to attach to surfaces and each other. Dragging them toward the desired location may require higher powered lasers. Quantum dots are moving relatively fast, so we may also need higher powered lasers to quicken the flows to collect them. But it should be noted that changing the laser power only changes the drag force, which affects the aggregate growth rate. In turn, growing the aggregate modifies back the flow fields. The laser is only a tool to control the intrinsic feedback between the aggregate and the flow fields.

Every experiment receives a different amount of external energy; the time to collect the particles and organisms differs from experiment to experiment, and characteristic length scales for each entity differ since the thermal bath temperature changes with the amount of injected energy, aggregate size, and the flow velocity. Neither these dynamics nor the vast number of degrees of freedom pose a problem, as the intrinsic feedback loops are in charge and we are in charge of the loops using only two control parameters: the laser power and beam position. Nonetheless, Schilling is correct: separation of energy, time, and length scales will be important when measuring the emergent dynamics of various patterns and behavior when one type of material is used. This concern is addressed in our latest work, which we are currently preparing for publication.

Among the other topics discussed in her letter, Schilling raises an important point when she stresses the necessity to generalize our self-assembly method to physically different systems operating far from equilibrium. The self-assembly method resulted from an evolving idea that my main collaborator, F. Ömer Ilday, and I have been working on for the better part of a decade. We have long appreciated the necessity of the triple mechanism of nonlinearity, fluctuations, and feedback loops in controlling dynamic adaptive systems. Accordingly, we investigated each pillar of the triple mechanism in entirely different experimental settings to understand their roles and the limits of their influence on self-assembly, self-organization, dynamic pattern formation, and adaptive behavior.¹² Some of these results have been published and others are on the way. We are consolidating all of these investigations into a mathematical framework, which we hope will bring a new perspective to the ongoing efforts to understand the principles of systems far from equilibrium.

Sailing Troubled Waters

Chemical systems have been the bedrock of self-assembly research. As Engel mentions, some fascinating results show a series of complex pattern formation and behavior. Engel then makes a critical point, noting that “highly interconnected systems, such as chemical reaction networks, are almost impossible to understand from first principles.” I wholeheartedly agree. Highly nonlinear and strongly stochastic settings constantly change the potential energy landscapes of systems far from equilibrium. Understanding or controlling this process is incredibly challenging. It feels like trying to chart a course in troubled waters. Not to mention that all solutions to gain control over a particular chemical system are system-specific and cannot be readily applied to other chemical systems.

Engel writes that “it appears there is no general extremum principle that governs the evolution of systems far from equilibrium.” Rolf Landauer arrived at a similar conclusion with his so-called blowtorch experiment, discussed partially in the 1970s and fully in 1988.¹³ But Landauer’s result argues against only locally calculated quantities in phase space and does not rule out an extremum principle that might consider the entire phase space. In her letter, Schilling disagrees with Engel and points to a new “power functional theory.”

Over the past several decades, advances in evolutionary biology have led to the development of a toolbox for navigating the changing energy landscapes via so-called fitness landscapes. Recent developments in artificial intelligence, graph theory, and genetic algorithms, to name a few, are continuously being added to this toolbox. Many other exciting developments along similar lines are emerging from diverse areas ranging from the social sciences to engineering and mathematics.

Purely physical experimental systems, similar to ours, are excellent candidates to approach far-from-equilibrium phenomena from first principles by automatically shrinking the parameter space to be controlled and significantly decreasing the degrees of freedom to be considered. I say this because I firmly believe that, to avoid getting lost in the whirlpools of chemical or biological complexity, the experimental systems should be as simple as possible, but no simpler—to paraphrase Albert Einstein. I also envision that the knowledge and capabilities gained from such simply complex systems can be readily translated to chemical and biological systems to guide their self-assembly and organization processes toward tailor-made dynamic adaptive topologies, functionalities, and behavior.

In closing, I would like to stress that our system is not exceptional or uniquely capable. Many other purely physical systems with settings and dynamics entirely different than ours should be established to study phenomena far from equilibrium, including the mode-locking of lasers, one of the earliest experimental systems considered by Haken and others in this context.¹⁴ In setting up such systems, two key aspects should be addressed: incorporating the triple mechanism of nonlinearity, fluctuations, and feedback mechanisms, and using an energy source that can deliver energy with a spatiotemporal precision. The more systems with insightful dynamics there are, the better.¹⁵

Serim Ilday et al.

Michael Engel replies:

In her letter, Serim Ilday writes that her response would address two of my criticisms. Here, I would like to substantiate those two rather brief comments to clarify why I believe that “[d]ue to the diversity of the systems studied by Makey et al., their data analysis presents some challenges.” Neither her response nor my email correspondence with Ilday during the summer of 2020 helped to address these concerns. This does not invalidate the impact of her team’s paper for Nature Physics.¹⁶ After all, as I wrote, it demonstrated by “clever utilization of the nonlinear multiphoton absorption of ultrafast pulses” a new model system for dissipative self-assembly across the scales. I hope the following comments may direct follow-up work toward a complete scientific understanding of the underlying processes.

1. “Filling area,” I wrote in my essay, “is not a convenient measure to study growth because it does not increase linearly, even in static self-assembly.” Nonlinearity is a characteristic of feedback. To first approximation, a growing two-dimensional nucleus—a static self-assembly process—adds material quadratically at first and then slows down as the coverage of the area nears complete coverage. My comment pointed out that the quantity the authors measured—filling ratio—is nonlinear and causes a sigmoidal curve even in the absence of feedback.
2. “The extraction of TW statistics,” I noted, “is reliant on high-pass filtering in the form of a temporal span analysis, [which] required choices of the temporal span parameter in a certain window and was performed for only a single particle type.” My comment repeated a statement from the Methods section of the Nature Physics paper: “Over a fairly broad range, from ~650 ms to ~850 ms, the normalized moments of the experimental data agree equally well with that of the TW GUE up to eighth normalized moments.”¹⁷ Full agreement is absent outside of that, in my view, rather narrow range as shown in Extended Data Figure 10. The temporal span parameter was chosen by the authors a posteriori at the time of data analysis. Similar data for other particle types to test universality of the TW statistics is not presented.