We present EPR-Net, a novel and effective deep learning approach that tackles a crucial challenge in biophysics: constructing potential landscapes for high-dimensional non-equilibrium steady-state systems. EPR-Net leverages a nice mathematical fact that the desired negative potential gradient is simply the orthogonal projection of the driving force of the underlying dynamics in a weighted inner-product space. Remarkably, our loss function has an intimate connection with the steady entropy production rate (EPR), enabling simultaneous landscape construction and EPR estimation. We introduce an enhanced learning strategy for systems with small noise, and extend our framework to include dimensionality reduction and the state-dependent diffusion coefficient case in a unified fashion. Comparative evaluations on benchmark problems demonstrate the superior accuracy, effectiveness and robustness of EPR-Net compared to existing methods. We apply our approach to challenging biophysical problems, such as an eight-dimensional (8D) limit cycle and a 52D multi-stability problem, which provide accurate solutions and interesting insights on constructed landscapes. With its versatility and power, EPR-Net offers a promising solution for diverse landscape construction problems in biophysics.

Supramolecular self-assembly is ubiquitous in living system and is usually controlled to proceed in time and space through sophisticated reaction-diffusion processes, underpinning various vital cellular functions. In this contribution, we demonstrate how spatiotemporal self-assembly of supramolecular hydrogels can be realized through a simple reaction-diffusion-mediated transient transduction of pH signal. In the reaction-diffusion system, a relatively faster diffusion of acid followed by delayed enzymatic production and diffusion of base from the opposite site enables a transient transduction of pH signal in the substrate. By coupling such reaction-diffusion system with pH-sensitive gelators, dynamic supramolecular hydrogels with tunable lifetimes are formed at defined locations. The hydrogel fibers show interesting dynamic growing behaviors under the regulation of transient pH signal, reminiscent of their biological counterpart. We further demonstrate a proof-of-concept application of the developed methodology for dynamic information encoding in a soft substrate. We envision that this work may provide a potent approach to enable transient transduction of various chemical signals for the construction of new colloidal materials with the capability to evolve their structures and functionalities in time and space.

The dynamical properties of many complex physical and biological systems can be quantified from the energy landscape theory. Previous approaches focused on estimating the transition rate from landscape reconstruction based on data. However, for general non-equilibrium systems (such as gene regulatory systems), both the energy landscape and the probability flux are important to determine the transition rate between attractors. In this work, we proposed a data-driven approach to estimate non-equilibrium transition rate, which combines the kernel density estimation and non-equilibrium transition rate theory. Our approach shows superior performance in estimating transition rate from data, compared with previous methods, due to the introduction of a nonparametric density estimation method and the new saddle point by considering the effects of flux. We demonstrate the practical validity of our approach by applying it to a simplified cell fate decision model and a high-dimensional stem cell differentiation model. Our approach can be applied to other biological and physical systems.

Biological systems possess unique non-equilibrium functions, maintaining tight manipulation of their surroundings through inter-communication of multiple components and self-regulatory capability organized over different length scales. However, most artificial materials are incapable of communicating and self-regulating behavior due to their limited number of component and direct responsive modes. Herein, a new integrated self-regulation system is developed utilizing stimuli-responsive polyelectrolyte capsules as building blocks. The combination of stimuli-responsive capsules and enzyme immobilized microgels is designed to mimic life systems and its programmable interactive communications and self-regulation behavior is demonstrated through communication-feedback mechanism. Polyelectrolyte capsules can sense changes of their surrounding, then start the internal communication by releasing energy-rich cargo mimicking the behavior of the cells. The microgel particles subsequently complete closed-loop communication through providing negative feedback on capsules by enzymatic reaction and actuating pH-regulation of the whole system. Different communication modes and pH-regulation behaviors could be achieved by adjusting spatial and kinetic conditions. Proposed intelligent system is highly customizable due to the wide selection of encapsulated cargos, stimuli-responsive blocks and reaction networks, and would have broad influences in areas ranging from medical implants that assist in stabilizing body functions to microreactor system that regulate catalytic reactions.

Aging phenomena have been observed in many non-equilibrium systems such as polymers and glasses, where physical properties depend on the waiting time between the starting time of observation and the time when the temperature is changed. The aging is classified into two types on the basis of the waiting time dependence of an instantaneous relaxation time: When the relaxation time is always an increasing function of the waiting time, the aging is called Type I and when it depends on the protocol of the temperature change, the aging is called Type II. Aging of a random walk in three dimensions is investigated when the free energy landscape controlling the jump rate responds to temperature change with a delay. It is shown that the intermediate scattering function of the random walk model exhibits Type II aging. It is also shown that the relaxation time of the free energy landscape can be deduced from the waiting time dependence of the instantaneous relaxation time.

The Nobel Prize in Chemistry was awarded in 1987 and 2016 for research in supramolecular chemistry on the \"development and use of molecules with structure-specific interactions of high selectivity\" and the \"design and production of molecular machines\", respectively. This confirmed the explosive development of supramolecular chemistry. In addition, attempts have been made in systems chemistry to embody the complex functions of living organisms as artificial non-equilibrium chemical systems, which have not received much attention in supramolecular chemistry. In this review, we explain recent developments in supramolecular chemistry through four categories: stimuli-responsiveness, time evolution, dissipative self-assembly, and hierarchical expression of functions. We discuss the development of non-equilibrium supramolecular systems, including the use of molecules with precisely designed properties, to achieve functions found in life as a hierarchical chemical system.

Amyloid fibrils are ordered, non-covalent polymers of proteins that are linked to a range of diseases, as well as biological functions. Amyloid fibrils are often considered thermodynamically so stable that they appear to be irreversible, explaining why very few quantitative thermodynamic studies have been performed on amyloid fibrils, compared to the very large body of kinetic studies. Here we explore the thermodynamics of amyloid fibril formation by the protein PI3K-SH3, which forms amyloid fibrils under acidic conditions. We use quartz crystal microbalance (QCM) and develop novel temperature perturbation experiments based on differential scanning fluorimetry (DSF) to measure the temperature dependence of the fibril growth and dissociation rates, allowing us to quantitatively describe the thermodynamic stability of PI3K-SH3 amyloid fibrils between 10 and 75°C.

Cell movement is crucial for morphogenesis in multicellular organisms. Growing embryos or tissues often expand isotropically, i.e., uniformly, in all dimensions. On the surfaces of these expanding environments, which we call \"fields,\" cells are subjected to frictional forces and move passively in response. However, the potential roles of isotropically expanding fields in morphogenetic events have not been investigated well. Our previous mathematical simulations showed that a tissue was elongated on an isotropically expanding field (Imuta et al., 2014). However, the underlying mechanism remains unclarified, and how cells behave during tissue elongation was not investigated. In this study, we mathematically analyzed the effect of isotropically expanding fields using a vertex model, a standard type of multi-cellular model. We found that cells located on fields were elongated along a similar direction each other and exhibited a columnar configuration with nearly single-cell width. Simultaneously, it was confirmed that the cell clusters were also elongated, even though field expansion was absolutely isotropic. We then investigated the mechanism underlying these counterintuitive phenomena. In particular, we asked whether the dynamics of elongation was predominantly determined by the properties of the field, the cell cluster, or both. Theoretical analyses involving simplification of the model revealed that cell clusters have an intrinsic ability to asymmetrically deform, leading to their elongation. Importantly, this ability is effective only under the non-equilibrium conditions provided by field expansion. This may explain the elongation of the notochord, located on the surface of the growing mouse embryo. We established the mechanism underlying tissue elongation induced by isotropically expanding external environments, and its involvement in collective cell alignment with cell elongation, providing key insight into morphogenesis involving multiple adjacent tissues.

In recent work we uncovered intriguing connections between Otto\'s characterization of diffusion as an entropic gradient flow on the one hand and large-deviation principles describing the microscopic picture (Brownian motion) on the other. In this paper, we sketch this connection, show how it generalizes to a wider class of systems and comment on consequences and implications. Specifically, we connect macroscopic gradient flows with large-deviation principles, and point out the potential of a bigger picture emerging: we indicate that, in some non-equilibrium situations, entropies and thermodynamic free energies can be derived via large-deviation principles. The approach advocated here is different from the established hydrodynamic limit passage but extends a link that is well known in the equilibrium situation.

The incoherent type-1 feed-forward loop (I1-FFL) is ubiquitous in biological regulatory circuits. Although much is known about the functions of the I1-FFL motif, the energy cost incurred in the network and how it affects the performance of the network have not been investigated. Here, we study a generic I1-FFL enzymatic reaction network modelled after the GEF-GAP-Ras pathway responsible for chemosensory adaptation in eukaryotic cells. Our analysis shows that the I1-FFL network always operates out of equilibrium. Continuous energy dissipation is necessary to drive an internal phosphorylation-dephosphorylation cycle that is crucial in achieving strong short-time response and accurate long-time adaptation. In particular, we show quantitatively that the energy dissipated in the I1-FFL network is used (i) to increase the system\'s initial response to the input signals; (ii) to enhance the adaptation accuracy at steady state; and (iii) to expand the range of such accurate adaptation. Moreover, we find that the energy dissipation rate, the catalytic speed and the maximum adaptation accuracy in the I1-FFL network satisfy the same energy-speed-accuracy relationship as in the negative-feedback-loop (NFL) networks. Because the I1-FFL and NFL are the only two basic network motifs that enable accurate adaptation, our results suggest that a universal cost-performance trade-off principle may underlie all cellular adaptation processes independent of the detailed biochemical circuit architecture.