Living matter produces a variety of beautiful spatiotemporal structures and patterns that are not enduringly present in their nonliving counterparts. These ordered, non-equilibrium steady states are often sustained through the consumption of energy. Here, we investigate the energetic cost of assembling an ordered aster from an initially disordered, uniform mixture of cytoskeletal microtubules and kinesin motors. Using a calibrated fluorescent ATP reporter, we measure reproducible radial ATP gradients on scales of tens of microns that establish within, and persist over, tens of minutes, alongside coupled spatial gradients in motor density. These appreciable gradients are predicted by a reaction-diffusion model that acknowledges the localization of ATP consumption to regions where both molecular motors and microtubules are sufficiently abundant to encourage consumption, as confirmed by finite element modeling. With our results, we compare the power per volume required by our cytoskeletal networks with the known power per volume expenditure in cells. Comparison of our measured results with estimates of the dissipative processes available to motor-microtubule mixtures leads to the hypothesis that maintaining spatial motor gradients dominates the energetic demand in this system. Our direct quantification of energetic fluxes across space unlocks future explorations of what steady states are accessible to cells, and how the cytoskeleton drives broad spatial organization.

While coding regions in the genome have a direct interpretation in terms of protein products, significant fractions are non-coding and yet control essential biological functions. Unlike the genetic code, there is no "lookup table" that identifies where regulatory proteins, known as transcription factors (TFs), bind. Here, we extract these binding sites by distilling sequences of nucleotide letters into collective coordinates (hyperletters) representing the binding sites that are active under specific environmental conditions. Going beyond local information footprints between individual bases and expression levels, our information blueprint algorithm compresses the global information by optimising filters that simultaneously scan an entire promoter sequence. Inspired by renormalisation-group techniques, we identify TF binding sites as coarse-grained variables combining groups of correlated mutations with the highest collective impact on gene expression. We validate our approach on experimental data for E. coli and discover novel regulatory elements illustrating its deployment at scale across growth conditions.

Genes are connected in complex networks of interactions where often the product of one gene is a transcription factor that alters the expression of another. Many of these networks are based on a few fundamental motifs leading to switches and oscillators of various kinds. And yet, there is more to the story than which transcription factors control these various circuits. These transcription factors are often themselves under the control of effector molecules that bind them and alter their level of activity. Traditionally, much beautiful work has shown how to think about the stability of the different states achieved by these fundamental regulatory architectures by examining how parameters such as transcription rates, degradation rates and dissociation constants tune the circuit, giving rise to behavior such as bistability. However, such studies explore dynamics without asking how these quantities are altered in real time in living cells as opposed to at the fingertips of the synthetic biologist's pipette or on the computational biologist's computer screen. In this paper, we make a departure from the conventional dynamical systems view of these regulatory motifs by using statistical mechanical models to focus on endogenous signaling knobs such as effector concentrations rather than on the convenient but more experimentally remote knobs such as dissociation constants, transcription rates and degradation rates that are often considered. We also contrast the traditional use of Hill functions to describe transcription factor binding with more detailed thermodynamic models. This approach provides insights into how biological parameters are tuned to control the stability of regulatory motifs in living cells, sometimes revealing quite a different picture than is found by using Hill functions and tuning circuit parameters by hand.

Microbial metabolism is impressively flexible, enabling growth even when available nutrients differ greatly from biomass in redox state. E. coli, for example, rearranges its physiology to grow on reduced and oxidized carbon sources through several forms of fermentation and respiration. To understand the limits on and evolutionary consequences of metabolic flexibility, we developed a mathematical model coupling redox chemistry with principles of cellular resource allocation. Our integrated model clarifies key phenomena, including demonstrating that autotrophs grow slower than heterotrophs because of constraints imposed by intracellular production of reduced carbon. Our model further indicates that growth is improved by adapting the redox state of biomass to nutrients, revealing an unexpected mode of evolution where proteins accumulate mutations benefiting organismal redox balance.

Cellular structures must organize themselves within strict physical constraints, operating with finite resources and well-defined boundaries. Classical systems demonstrate only passive responses to boundaries, from surface energy minimization in soap films to strain distributions in elastic networks. Active matter fundamentally alters this paradigm - internally generated stresses create a bidirectional coupling between boundary geometry and mass conservation that enables dynamic control over network organization. Here we demonstrate boundary geometry actively directs network deformation in reconstituted microtubule-kinesin systems, revealing a programmable regime of shape transformation through controlled boundary manipulation. A coarse-grained theoretical framework reveals how boundary geometry couples to internal stress fields via mass conservation, producing distinct dynamical modes that enable engineered deformations. The emergence of shape-preserving and shape-changing regimes, predicted by theory and confirmed through experiments, establishes boundary geometry as a fundamental control parameter for active materials. The control principle based on boundaries advances both the understanding of biological organization and enables design of synthetic active matter devices with programmable deformation.

Many biological processes can be thought of as the result of an underlying dynamics in which the system repeatedly undergoes distinct and abortive trajectories with the dynamical process only ending when some specific process, purpose, structure or function is achieved. A classic example is the way in which microtubules attach to kinetochores as a prerequisite for chromosome segregation and cell division. In this example, the dynamics is characterized by apparently futile time histories in which microtubules repeatedly grow and shrink without chromosomal attachment. We hypothesize that for biological processes for which it is not the initial conditions that matter, but rather the final state, this kind of exploratory dynamics is biology's unique and necessary solution to achieving these functions with high fidelity. This kind of cause and effect relationship can be contrasted to examples from physics and chemistry where the initial conditions determine the outcome. In this paper, we examine the similarities of many biological processes that depend upon random trajectories starting from the initial state and the selection of subsets of these trajectories to achieve some desired functional final state. We begin by reviewing the long history of the principles of dynamics, first in the context of physics, and then in the context of the study of life. These ideas are then stacked up against the broad categories of biological phenomenology that exhibit exploratory dynamics. We then build on earlier work by making a quantitative examination of a succession of increasingly sophisticated models for exploratory dynamics, all of which share the common feature of being a series of repeated trials that ultimately end in a "winning" trajectory. We also explore the ways in which microscopic parameters can be tuned to alter exploratory dynamics as well as the energetic burden of performing such processes. It is a great privilege to take part in this special volume dedicated to the life and work of Prof. Erich Sackmann (1934-2024). For one of us (RP), at the time of making a switch from traditional condensed matter physics to a life engaged in the study of life, he went to a meeting near Munich which completely opened his eyes to the ways in which the approach of physics could be brought to bear on the study of the living. Sackmann's work was an inspiring presence at that meeting. One of the hallmarks of his work was a principled approach to dissecting biological processes over a range of scales and phenomena. One common thread to much of his work was that it acknowledged the dynamical character of living organisms. The present paper attempts to follow in the tradition of Sackmann's studies of dynamics by suggesting a new way of looking at many biological processes all through the unifying perspective of what we will call exploratory dynamics.

Cells control fluid flows with a spatial and temporal precision that far exceeds the capabilities of current microfluidic technologies. Cells achieve this superior spatio-temporal control by harnessing dynamic networks of cytoskeleton and motor proteins. Thus, engineering systems to mimic cytoskeletal protein networks could lead to the development of a new, active-matter-powered microfluidic device with improved performance over the existing technologies. However, reconstituted motor-microtubule systems conventionally generate chaotic flows and cannot perform useful tasks. Here, we develop an all-optical platform for programming flow fields for transport, separation and mixing of cells and particles using networks of microtubules and motor proteins reconstituted in vitro. We employ mathematical modeling for design optimization, which enables the construction of flow fields that achieves micron-scale transport. We use the platform to demonstrate that active-matter-generated flow fields can probe the extensional rheology of polymers, such as DNA, achieve transport and mixing of beads and human cells, and isolation of human cell clusters. Our findings provide a bio-inspired pathway for programmatically engineering dynamic micron-scale flows and demonstrate the vast potential of active matter systems as an engineering technology.

Is a herd of wildebeest better thought of as a series of individual animals, each with its own glorious and unmanageable volition, or as a field of moving arrows? Are the morphogen gradients that set up the coordinate systems for embryonic anterior-posterior patterning a smooth and continuous concentration field or instead a chaotic collection of protein molecules each jiggling about in the haphazard way first described by Robert Brown in his microscopical observations of pollen? Is water, the great liquid ether of the living world, a collection of discrete molecules or instead a perfectly continuous medium with a density of ≈1000 kg/m3? In this article, I will argue that these questions pose a false dichotomy since there are many different and powerful representations of the world around us. Different representations suit us differently at different times and it is often useful to be able to hold these seemingly contradictory notions in our heads simultaneously. Indeed, mathematics is not only the language of representation, but often is also the engine of reconciliation of such disparate views. In a letter to Alfred Russel Wallace on 14 April 1869, Charles Darwin noted that Lord Kelvin's "views on the recent age of the world have been for some time one of my sorest troubles". Here, I will argue that one of the highest attainments of the scientific enterprise is a coherent picture of the world, a picture in which our stories about the geological age of the Earth are coherent with our stories of how whales populated the oceans, our understanding of the living jibes with our understanding of the inanimate, our insights into the dynamics of genes and molecular structures are consonant with our physical understanding of the laws of statistical physics. The underpinnings of such coherency are often best revealed when viewed through the lens of mathematics.

A hallmark feature of active matter systems is the ability of individual elements to interact and organize over length scales exceeding that of the constituent molecular players. However, the nature of internal redistribution that occurs in the bulk of the collective is less clear. Using light-dimerizable kinesin motors to spatially control the formation and contraction of a microtubule network, we deliberately photobleach a grid pattern onto the filament network serving as a transient and dynamic coordinate system to observe the deformation and translation of the remaining fluorescent squares of microtubules. We find that the network contracts at a rate set by motor speed but is accompanied by a diffusive-like spread throughout the bulk of the contracting network with effective diffusion constant two orders of magnitude lower than that for freely diffusing microtubules. We further find that on micron scales, the diffusive timescale is only a factor of ≈3 slower than that of advection regardless of conditions, showing that the global contraction and long-time relaxation from this diffusive behavior are both motor-driven but exhibit local competition within the network bulk.

All cells respond to changes in both their internal milieu and the environment around them through the regulation of their genes. Despite decades of effort, there remain huge gaps in our knowledge of both the function of many genes (the so-called y-ome) and how they adapt to changing environments via regulation. Here we describe a joint experimental and theoretical dissection of the regulation of a broad array of over 100 biologically interesting genes in E. coli across 39 diverse environments, permitting us to discover the binding sites and transcription factors that mediate regulatory control. Using a combination of mutagenesis, massively parallel reporter assays, mass spectrometry and tools from information theory and statistical physics, we go from complete ignorance of a promoter's environment-dependent regulatory architecture to predictive models of its behavior. As a proof of principle of the biological insights to be gained from such a study, we chose a combination of genes from the y-ome, toxin-antitoxin pairs, and genes hypothesized to be part of regulatory modules; in all cases, we discovered a host of new insights into their underlying regulatory landscape and resulting biological function.

Control of transcription presides over a vast array of biological processes including through gene regulatory circuits that exhibit multistability. Two- and three-gene network motifs are often found to be critical parts of the repertoire of metabolic and developmental pathways. Theoretical models of these circuits, however, typically vary parameters such as dissociation constants, transcription rates, and degradation rates without specifying precisely how these parameters are controlled biologically. In this paper, we examine the role of effector molecules, which can alter the concentrations of the active transcription factors that control regulation, and are ubiquitous to regulatory processes across biological settings. We specifically consider allosteric regulation in the context of extending the standard bistable switch to three-gene networks, and explore the rich multistable dynamics exhibited in these architectures as a function of effector concentrations. We then study how the conditions required for tristability and more complex dynamics, and the bifurcations in dynamic phase space upon tuning effector concentrations, evolve under various interpretations of regulatory circuit mechanics, the underlying activity of inducers, and perturbations thereof. Notably, the biological mechanism by which we model effector control over dual-function proteins transforms not only the phenotypic trend of dynamic tuning but also the set of available dynamic regimes. In this way, we determine key parameters and regulatory features that drive phenotypic decisions, and offer an experimentally tunable structure for encoding inducible multistable behavior arising from both single and dual-function allosteric transcription factors.

Glycolysis is a conserved metabolic pathway that produces ATP and biosynthetic precursors. It is not well understood how the control of mammalian glycolytic enzymes through allosteric feedback and mass action accomplishes various tasks of ATP homeostasis, such as controlling the rate of ATP production, maintaining high and stable ATP levels, ensuring that ATP hydrolysis generates a net excess of energy, and maintaining glycolytic intermediate concentrations within physiological levels. To investigate these questions, we developed a biophysical model of glycolysis based on enzyme rate equations derived from in vitro kinetic data. This is the first biophysical model of human glycolysis that successfully recapitulates the above tasks of ATP homeostasis and predicts absolute concentrations of glycolytic intermediates and isotope tracing kinetics that align with experimental measurements in human cells. We use the model to show that mass action alone is sufficient to control the ATP production rate and maintain the high energy of ATP hydrolysis. Meanwhile, allosteric regulation of hexokinase and phosphofructokinase by ATP, ADP, inorganic phosphate, and glucose-6-phosphate is required to maintain high ATP levels and to prevent uncontrolled accumulation of phosphorylated intermediates of glycolysis. Allosteric feedback achieves the latter by maintaining hexokinase and phosphofructokinase enzyme activity at one-half of ATP demand and, thus, inhibiting the reaction of Harden and Young, which otherwise converts glucose to supraphysiological levels of phosphorylated glycolytic intermediates at the expense of ATP. Our methodology provides a roadmap for a quantitative understanding of how metabolic homeostasis emerges from the activities of individual enzymes.

A variety of biomolecular systems rely on exploratory dynamics to reach target locations or states within a cell. Without a mechanism to remotely sense and move directly toward a target, the system must sample over many paths, often including resetting transitions back to the origin. We investigate how exploratory dynamics can confer an important functional benefit: the ability to respond to small changes in parameters with large shifts in the steady-state behavior. However, such enhanced sensitivity comes at a cost: resetting cycles require energy dissipation to push the system out of its equilibrium steady state. We focus on minimalist models for two concrete examples: translational proofreading in the ribosome and microtubule length control via dynamic instability to illustrate the trade-offs between energetic cost and sensitivity. In the former, a driven hydrolysis step enhances the ability to distinguish between substrates and decoys with small binding energy differences. In the latter, resetting cycles enable catalytic control, with the steady-state length distribution modulated by substoichiometric concentrations of a reusable catalyst. Synthesizing past models of these well-studied systems, we show how path-counting and circuit-mapping approaches can be used to address fundamental questions such as the number of futile cycles inherent in translation and the steady-state length distribution of a dynamically unstable polymer. In both cases, a limited amount of thermodynamic driving is sufficient to yield a qualitative transition to a system with enhanced sensitivity, enabling accurate discrimination and catalytic control at a modest energetic cost.

Life uses non-equilibrium mechanisms to create ordered structures not attainable at equilibrium; the resulting order is assumed to provide functional benefits that outweigh costs of time and energy needed by these mechanisms. Here, we show that models of DNA replication and self-assembly, when expanded to include known stalling effects, can evolve error correcting mechanisms like kinetic proofreading and dynamic instability through selection for fast replication alone. We abstract these results into a general framework that predicts a counterintuitive ''order through speed'' effect if the distribution of replication times is wide enough. We test our results against recent mutational screens of proofreading polymerases. Our work suggests the intriguing possibility that non-equilibrium order can evolve even before that order is directly functional, with consequences for the evolution of mutation rates, viral capsid assembly, and the origin of life.

Living cells perform tasks by generating mechanical force through networks of protein filaments that are actively organized by molecular motors. Eukaryotic genomes encode diverse motor protein variants with distinct biochemical properties, which cells deploy for specific processes such as spindle formation during division or cilia for navigation. Despite extensive characterization of motor–filament systems by in vitro and in vivo approaches, how variations in genomic sequence and structure translate into filament organization dynamics remains poorly understood. Here we develop ActiveDROPS, a cell-free approach to reconstitute microtubule dynamics driven by genetically encoded kinesin motor variants in bacterial lysate droplets, enabling quantitative characterization of motor-induced cytoskeletal behavior in a cell-like context. We screened a library of uncharacterized kinesin-like sequences and identified 12 functional motors that can be grouped into three behavioral classes: “slow-sustained” (prolonged motion > 12 h, velocity 10–50 nm/s), “fast-burst” (brief motion < 2 h, velocity 50– 1000 nm/s), and “multiphase” defined by concentration-dependent transitions including nematic flows, chiral rotation, and global contraction. Through modular recombination of domains from “slow-sustained” and “fast-burst” motors, we engineered chimeras with combined “fast-sustained” dynamics (duration > 12 h, velocity > 100 nm/s). This revealed domain sets controlling microtubule motion lifespan and velocity, with computational analysis of AlphaFold-predicted structures suggesting 0 and L12 as the respective determinants. Incorporation of microtubule-associated proteins such as PRC1 and Tau, together with motor co-expression, further broadened the range of cytoskeletal dynamics. These results provide a framework to reverse-engineer how nature generates mechanical diversity, offering insight into how cells transform genetic variation into emergent behavior. ### Competing Interest Statement The authors have declared no competing interest.

Cytoskeletal networks can repair defects to maintain structural integrity. However, the mechanisms and dynamics of defect merging remain poorly understood. Here we report a geometry-tunable merging mechanism in microtubule-motor networks initiated by active crosslinking. We directly generate defects using a light-controlled microtubule-motor system in O-shaped and V-shaped networks, and observe that the defects can self-close. Combining theory and experiment, we find that the V-shaped networks must overcome internal elastic resistance in order to zip up cracks, giving rise to a bifurcation of dynamics dependent on the initial opening angle of the crack: the crack merges below a critical angle and opens up at larger angles. Simulation of a continuum model reproduces the bifurcation dynamics, revealing the importance of overlapping boundary layers where free motors and microtubules can actively crosslink and thereby merge the defects. We also formulate a simple elastic-rod model that can qualitatively predict the critical angle, which is tunable by the network geometry.

In biological systems, molecular-scale forces and motions are pivotal for enabling processes like motility, shape change, and replication. These forces and motions are organized, amplified, and transmitted across macroscopic scales by active materials such as the cytoskeleton, which drives micron-scale cellular movement and re-organization. Despite the integral role of active materials, understanding how molecular-scale interactions alter macroscopic structure and force propagation remains elusive. This knowledge gap presents challenges to the harnessing and regulation of such dynamics across diverse length scales. Here, we demonstrate how mediating the bundling of microtubules can shift active matter between a global force-transmitting phase and a local force-dissipating phase. A fivefold increase in microtubule effective length results in the transition from local to global phase with a hundredfold increase in velocity autocorrelation. Through theory and simulation, we identify signatures of a percolation-driven transition between the two phases. This provides evidence for how force propagation can be generated when local molecular interactions reach a sufficient length scale. We show that force propagation in the active matter system enables material transport. Consequently, we demonstrate that the global phase is capable of facilitating millimeter-scale human cell transport and manipulation, as well as powering the movement of aqueous droplets. These findings underscore the potential for designing active materials capable of force organization and transmission. Our results lay the foundation for further exploration into the organization and propagation of forces/stresses in biological systems, thereby paving the way for the engineering of active materials in synthetic biology and soft robotics.

For the vast majority of genes in sequenced genomes, there is limited understanding of how they are regulated. Without such knowledge, it is not possible to perform a quantitative theory-experiment dialogue on how such genes give rise to physiological and evolutionary adaptation. One category of high-throughput experiments used to understand the sequence-phenotype relationship of the transcriptome is massively parallel reporter assays (MPRAs). However, to improve the versatility and scalability of MPRA pipelines, we need a "theory of the experiment" to help us better understand the impact of various biological and experimental parameters on the interpretation of experimental data. These parameters include binding site copy number, where a large number of specific binding sites may titrate away transcription factors, as well as the presence of overlapping binding sites, which may affect analysis of the degree of mutual dependence between mutations in the regulatory region and expression levels. To that end, in this paper we create tens of thousands of synthetic single-cell gene expression outputs using both equilibrium and out-of-equilibrium models. These models make it possible to imitate the summary statistics (information footprints and expression shift matrices) used to characterize the output of MPRAs and from this summary statistic to infer the underlying regulatory architecture. Specifically, we use a more refined implementation of the so-called thermodynamic models in which the binding energies of each sequence variant are derived from energy matrices. Our simulations reveal important effects of the parameters on MPRA data and we demonstrate our ability to optimize MPRA experimental designs with the goal of generating thermodynamic models of the transcriptome with base-pair specificity. Further, this approach makes it possible to carefully examine the mapping between mutations in binding sites and their corresponding expression profiles, a tool useful not only for better designing MPRAs, but also for exploring regulatory evolution.