Rheology of Glass and Jamming

— Uncovering the Physical Commonality Between Mayonnaise and Glass —

(Collaborative research with the group of Prof. Masashi Ikeda, The University of Tokyo)

We have demonstrated that the unusual viscoelastic properties observed in “soft jammed solids,” such as mayonnaise and shaving foam, are deeply related to characteristic vibrational modes found in glasses and amorphous solids—namely, quasi-localized vibrations known as the boson peak.

Soft jammed solids are materials formed by densely packed, soft particles arranged in a disordered manner, exhibiting properties intermediate between liquids and solids. Although they are widely used in industrial products and are abundant in everyday life, their fundamental mechanical propertieshave not been fully understood. In particular, the phenomenon of anomalously large viscous dissipationunder slowly applied external forces has long remained unexplained within conventional viscoelastic theories.

To address this problem, we performed viscoelastic measurements of dense emulsions using microrheology. By employing optical techniques, we precisely measured the thermal fluctuations of embedded probe particles with high spatiotemporal resolution. This approach enabled us to quantitatively capture the anomalous viscoelastic response of soft jammed solids with unprecedented precision (Fig. 1).
A key finding is the clear scaling law in which theviscous loss increases proportionally to the square root of frequency (∝ ω¹ᐟ²). Such non-standard frequency dependence does not appear in ordinary viscoelastic materials and is considered a hallmark of disordered structures.

Through detailed theoretical analysis of the experimental data, we found that the origin of this phenomenon lies in dissipative vibrational modes associated with the random contact network between particles—corresponding to the boson peak. In glasses, vibrational modes propagating through disordered contact networks include quasi-localized and non-elastic components that lead to energy dissipation. When subjected to an external oscillatory force, these modes respond in a way that prevents energy storage and instead promotes dissipation.
In soft jammed solids, anomalous relaxation modes emerge that correspond to these glassy vibrational modes (Fig. 2), and they quantitatively reproduce the observed ω¹ᐟ² scaling (Fig. 1).

This relationship is schematically illustrated in Fig. 2 of our study. Understanding how structural disorder within soft jammed solids gives rise to such anomalous relaxation modes—and how these modes dissipate energy in response to external forces—is both an intriguing and challenging problem. By clarifying the connection between glasses and soft jammed solids, our work provides an intuitive framework for understanding the mechanism that bridges macroscopic forcing and microscopic structural fluctuations through energy dissipation.

This study represents one of the first demonstrations that soft matter systems such as “softly jammed materials” and amorphous solids such as disordered glasses can be described within a unified framework of condensed matter physics. These findings are expected to provide new insights into the physics of nonequilibrium materials and to guide the design of next-generation materials.

This work has been published in Nature Physics.
DOI: 10.1038/s41567-024-02722-7

Non-equilibrium Rheology of Cells

Development of a Method to Directly Measure Nonequilibrium Rheology Inside Living Cells

We have successfully developed a method to measure the rheological properties of the interior of living cells—namely stiffness, viscosity, and the magnitude of forces generated by the cell—which had long been considered technically inaccessible.

Rheology describes how materials flow and deform, and it is an essential property for evaluating the performance and durability of industrial products and advanced technologies. Similarly, to properly understand the function and state of cells—the fundamental units of living organisms—it is highly desirable to measure their rheological properties. However, cells are extremely small (approximately 10 micrometers in diameter), and their interiors are continuously driven by active processes powered by molecular-scale motors. Accurately measuring material properties in such a small and dynamically active environment has therefore been a major challenge. Until now, most approaches relied on probing the exterior of cells—effectively “scratching” the surface—to infer properties near the membrane, making it impossible to directly access the true internal state of the cell.

To overcome this limitation, we developed a method in which microscopic probe particles are introduced into living cells, and their fluctuations and responses to externally applied forces are measuredwith high precision. Inside cells, vigorous motion known as cytoplasmic streaming is constantly present. By employing feedback control, we were able to compensate for this motion in real time and track the position of the probe particles with sub-nanometer accuracy (Fig. 3).

It is important to note that, in living cells, probe particles are actively driven by the cell itself. Therefore, large fluctuations of the particles do not necessarily indicate that the cell is soft or fluid-like (this interpretation is only valid in dead cells). To address this, we measured the response of the particles to controlled forces applied by laser trapping, allowing us to accurately extract the intrinsic rheological properties of the cellular interior. Furthermore, by simultaneously measuring both the fluctuations and the response of the particles, we were able to quantify how much force the cell generates internally—effectively how “active” or “energetic” the cell is—in terms of the violation of the fluctuation–dissipation theorem (Fig. 4).

The measurement technique we have developed thus provides the first direct evaluation of the physical properties inside living cells. It is expected to make significant contributions across a wide range of fields, including physics, cell biology, and medicine. For further details, please refer to ourreview articles and original publications.

Cytoplasm as Active Glass

Crowding and “Stirring” Generate the Flexibility of Living Cells

We compared the viscoelastic properties (stiffness and viscosity) of living cells with those of cell extracts, in which the cellular contents are removed from their native context, in order to identify the origin of their differences. Our results revealed that the combination of intracellular crowding and the stirring activity of motor proteinsplays a decisive role (Fig. 5).

Cells can dynamically regulate their mechanical properties—such as stiffness and viscosity—according to functional demands. In contrast, for inanimate materials such as glasses or gels, altering their properties generally requires restructuring the material itself. Living cells, however, can modulate their properties in a far more flexible and dynamic manner. This remarkable “flexibility” arises from the crowded intracellular environment and the active stirring driven by motor proteins. If these factors strongly influence the fluidity and viscosity of the material, then cells may indeed possess fundamentally “lifelike” properties that distinguish them from ordinary matter.

Without Crowding and Stirring, Cells Solidify

We first investigated cell extracts—prepared by disrupting the cell membrane and isolating the intracellular contents—in which the stirring activity of motor proteins is effectively removed. By varying the concentration of the contents, we measured their mechanical properties under these conditions. We found that evena modest increase in concentration led to a dramatic rise in viscosity, eventually causing the entire extract to solidify (Fig.6).
Remarkably, this phenomenon was observed across a wide range of cell types, including human cells, bacteria, egg cells, and tissue cells. Furthermore, solidification began at concentrations lower than those typically found inside living cells (approximately 300 mg/mL). These results suggest that,in the absence of active stirring, cellular contents would readily become solid-like. Such solidification would prevent essential processes such as molecular synthesis and transport, ultimately impairing cellular function.

Why Do Living Cells Not Solidify?

To address this question, we measured the viscoelastic properties of living cellswhile similarly varying the intracellular concentration. Surprisingly, despite having comparable concentrations, living cells maintained fluidity. Moreover, the manner in which viscosity changed with concentration was entirely different from that observed in cell extracts. Detailed analysis revealed that this difference originates from the cell’s intrinsic ability to actively “stir” its interior. In other words, cells prevent crowding-induced solidification through the activity of motor proteins.

Until recently, intracellular crowding and active stirring had not been central considerations in understanding cellular mechanics and function. However, our findings suggest that these factors are crucial for cellular flexibility, responsiveness, and potentially even cellular health. The mechanical properties of cells and tissues are known to influence a wide range of physiological and pathological processes, including cancer progression, development, reproduction, and stem cell differentiation. Our results represent an important step toward a deeper understanding of such phenomena and may open new avenues for applications across diverse fields. For further details, please refer to the original publication.

Dynamics of Non-Gaussian Fluctuation

— Constructing a New Theoretical Framework for Statistics Unique to Nonequilibrium Systems —

Background: Non-Gaussian Fluctuations in Nonequilibrium Systems

In physical systems at equilibrium, fluctuations of observables are generally expected to follow a Gaussian (normal) distribution, as guaranteed by the central limit theorem. In particular, at the mesoscale—intermediate between microscopic and macroscopic regimes—where systems can often be approximated as homogeneous continua, measured quantities should exhibit Gaussian statistics.
However, in real nonequilibrium systems, this expectation is frequently violated, and clearly non-Gaussian fluctuations are observed. Representative examples include turbulence, glassy and jammed systems, intracellular force generation and molecular transport, and suspensions of motile microorganisms (active matter). By elucidating the origin of such non-Gaussian distributions, we can gain deeper insight into the physical properties and dynamics of nonequilibrium systems through the shape of the distributions and their temporal evolution.

Construction of a New Non-Gaussian Limiting Distribution

In conventional statistical mechanics, when many independent contributions with finite variance are superposed, the central limit theorem ensures convergence to a Gaussian distribution. In contrast, when the system is dominated by interactions with heavy-tailed distributions whose variance diverges, the resulting distribution converges to a non-Gaussian stable distribution known as a Lévy distribution. Such statistical limits naturally arise in systems governed by long-range, power-law interactions, which are ubiquitous in nature. For example, gravitational interactions between stars in space, electrostatic forces in plasmas, and hydrodynamic interactions among motile microorganisms all decay as the inverse square of distance. Near each interaction source (e.g., particles, charges, or microorganisms), the interaction strength diverges, leading to divergent variance when measured at a mathematical point. In practice, however, measurements are always performed over finite regions, and the variance remains finite.

This raises the following question: when a large number of interaction sources are randomly distributed in three-dimensional space, what statistical distribution emerges from the superposition of their interactions? If the singular nature of individual interactions dominates, a Lévy distribution is expected; if finite-size effects dominate, a Gaussian distribution should arise. While the Gaussian approximation is often valid for equilibrium systems, nonequilibrium systems frequently exhibit distributions that are neither Gaussian nor Lévy.

To address this, we derived a new analytical expression for the limiting distribution of fluctuations arising from interactions generated by randomly distributed sources in three-dimensional space (e.g., motile microorganisms) (see original paper).

The characteristic function (Fourier transform) of this distribution is parameterized by:
• the characteristic system size R,
• the concentration of interaction sources c, and
• the interaction strength \gamma.

This formulation defines a new family of distributions that continuously interpolates between Gaussian and Lévy distributions. Although we present the explicit form for three dimensions here, a notable feature is that the properties of the distribution depend on the spatial dimensionality.

Connection to Real Nonequilibrium Fluctuations and Outlook

This newly derived non-Gaussian distribution has the potential to quantitatively describe fluctuations observed in a wide range of nonequilibrium systems, including:
1. suspensions of motile microorganisms (active matter),
2. actin–myosin gels,
3. glassy soft matter, and
4. systems exhibiting turbulence or jamming.

We are currently combining experiments, theoretical analysis, and numerical simulations to verify that this distribution accurately reproduces fluctuations observed in real systems. For case (1), experimental validation has already been completed and reported in our original publication. Furthermore, this theoretical framework can be extended to incorporate the temporal dynamics of the interaction sources themselves (e.g., microorganisms or molecular motors), providing a pathway toward a more realistic and comprehensive understanding of nonequilibrium dynamics.

Fluctuations and Energetics of Biomolecular Machines

Collaborative research with Dr. Takayuki Ariga (Osaka University)

Using an optical tweezers system equipped with high-speed feedback control, we have experimentally quantified the energy input and output at the single-molecule level for the walking biomolecular motor protein kinesin (Fig.7). Through mathematical modeling and theoretical analysis, we found that a large fraction of the chemical energy supplied to kinesin is not used for cargo transport but is instead dissipated as heat within the molecule.

Furthermore, we discovered that kinesin can be accelerated by applying artificially fluctuating forces that mimic the intracellular environment (Fig.8). Notably, this acceleration becomes more pronounced under high load conditions, suggesting that kinesin may be adapted to operate efficiently in crowded and highly viscous cellular environments. These findings imply that the non-thermal fluctuations present inside cells are not merely noise, but may be actively utilized to enhance physiological functions (see original paper).

Nonequilibrium Mechanics Using an Exchange Chamber

Probing the Effects of Metabolic Activity on Intracellular Mechanical Environments

Inside living cells, the mechanical environment is shaped by the dynamics of biomolecules that consume energy carriers such as ATP. Recent studies suggest that, although the intracellular environment is highly crowded and tends toward a glass-like state, metabolic activity enables it to maintain fluidity. This raises key questions:
• How does metabolic activity influence the dynamic mechanical environment inside cells?
• What are the molecular origins of intracellular fluidity?
Directly addressing these questions in living cells is challenging, because cells exhibit feedback responses that maintain homeostasis. Attempts to externally manipulate metabolic or mechanical conditions often trigger cellular responses, obscuring the underlying mechanisms.

A New Approach Using Cell Extracts and an Exchange Chamber

To overcome this limitation, we have developed a novel experimental system:
• Using cell extracts (intracellular contents isolated from cells), into which metabolic activity can be artificially introduced.
• Allowing independent control of molecular concentration and metabolic activity.

However, in conventional closed systems, metabolic activity cannot be sustained over long periodsdue to depletion of active components and accumulation of waste products. To address this, we developed an exchange chamber with the following features:
• Continuous supply of active molecules (e.g., ATP) through a semipermeable membrane
• Simultaneous removal of metabolic byproducts

This setup enables long-term microrheology measurements under sustained metabolic activity (Fig. 9). Using this system, we can directly investigate the effects of metabolism on intracellular mechanical properties without interference from cellular feedback mechanisms.

 

Bacterial Suspensions as a Model Nonequilibrium System

As a model system mimicking the nonequilibrium intracellular environment, we also study dense suspensions of Escherichia coli (Fig.10).

E. coli bacteria propel themselves by rotating flagella, driven by molecular motors that are not directly observable under a microscope. By treating the bacteria themselves as “visible motors,” we can effectively visualize the stirring effects generated by otherwise invisible molecular processes inside cells. At high concentrations, interactions among bacteria give rise to collective swirling flows known as bacterial turbulence. Such systems belong to a broader class known as active matter, which consists of self-driven particles that consume energy to generate motion. Active matter has become a major topic of research in both physics and biology.

Long-Term, Three-Dimensional Observation of Active Matter

Most previous studies of active matter have been limited to quasi-two-dimensional systems, where motion can be observed but is short-lived due to depletion of energy sources. However, mechanical measurements require three-dimensional systems, which have been difficult to realize. To overcome this, we constructed a three-dimensional active matter system using the exchange chamber. By continuously supplying nutrients and removing waste products through a semipermeable membrane, we achieved stable, long-term self-propelled motion of bacteria. Although three-dimensional observation is more challenging, we have confirmed the presence of sustained bacterial turbulence in this system (Movie 1,2).

 

Transition to an Active Glass State

At even higher bacterial concentrations, collective motion (swimming and turbulence) gradually disappears, and rearrangements between bacteria become suppressed. This state is known as an active glass, a quasi-arrested state in which energy is continuously supplied (through flagellar rotation), yet motion becomes localized and structural rearrangements are effectively frozen. This active glass state provides a useful model for understanding how mechanical properties emerge under conditions of high density, sustained activity, and nonequilibrium dynamics—conditions analogous to those inside living cells.

By combining the exchange chamber with bacterial suspensions, we have established a powerful experimental platform for microrheology in metabolically driven nonequilibrium environments. This system enables quantitative investigation of:
• how cells and biomolecules generate nonequilibrium dynamics, and
• how these dynamics influence mechanical properties such as viscoelasticity and fluidity.

Microrheology Using Optical Lever Detection

To measure the mechanical properties of polymer networks such as gels using microrheology, probe particles larger than the mesh size of the network are required. However, in laser interferometry-based measurements, sensitivity decreases when the probe size exceeds the optical wavelength (geometric optics regime).
To address this issue, we have designed an optical system using an optical lever, which maintains high sensitivity even for larger probe particles (Fig.11,12). In this method, the transmitted laser beam is collimated by the probe particle itself, effectively increasing the optical path length and enhancing displacement sensitivity. In addition, by incorporating adaptive optics, we aim to achieve high measurement sensitivity even in optically heterogeneous samples.

Microrheology of Dense Colloidal Suspensions Under Shear

When macroscopic shear is applied to glassy systems, it is well known that the viscosity exhibits nonlinear dependence on the applied stress. However, the microscopic mechanisms underlying this nonlinear response remain unclear.
In our laboratory, we aim to elucidate these mechanisms by measuring mechanical responses at the level of individual constituent particles under applied macroscopic shear (Fig.13). In parallel, we also perform measurements of microscopic mechanical responses to localized forces applied via optical trapping, allowing direct comparison between macroscopic and microscopic nonlinear dynamics.
We further complement these experiments with Brownian dynamics simulations near the glass transition to deepen our understanding of these phenomena (Movie 3).