Appetizer on soft matter physics concepts in mechanobiology

Abstract

Mechanosensing, the active responses of cells to the mechanics on multiple scales, plays an indispensable role in regulating cell behaviors and determining the fate of biological entities such as tissues and organs. Here, I aim to give a pedagogical illustration of the fundamental concepts of soft matter physics that aid in understanding biomechanical phenomena from the scale of tissues to proteins. Examples of up‐to‐date research are introduced to elaborate these concepts. Challenges in applying physics models to biology have also been discussed for biologists and physicists to meet in the field of mechanobiology.

1. MECHANOBIOLOGY: BIOLOGICAL RESPONSES TO MECHANICS

Life is an active machinery orchestrated by components of multiple scales in hierarchical structures. They are active in the sense that they adapt to turbulent environments for better and longer living. Such active responses exist not only on a large scale of the entire animal body in which their brains sense signals from the surroundings and make decisions, but also on cellular and even subcellular scales. Chemical signals have been the mainstream research target in bioactivity below cellular scales. However, it has been widely accepted that mechanical signals also play pivotal roles (Bershadsky et al., 2006; Geiger & Bershadsky, 2002; Matthews et al., 2006). Mechanobiology is the specific field encompassing the research at the merging edge of biology and physics.

Mechanical forces are ubiquitous in biological systems (Discher et al., 2005; Holle & Engler, 2011; Janmey & Weitz, 2004; Vogel & Sheetz, 2006). The bioentities, (e.g., organs, tissues, cells, nuclei, proteins, etc.) on extra‐, inter‐, and subcellular levels could “sense” force variations in space and time through the deformation and dynamics of themselves, which triggers the change in the machinery of energy and mass exchange during the biological processes (Cho et al., 2017; Sears & Kaunas, 2016; Smith et al., 2018; Tark‐Dame et al., 2011; Trichet et al., 2012).

The total scope of mechanobiology can be further divided into two subdivisions. First, on the mechanical side: what kind of mechanical information bioentities encounter, where, and how? Second, on the biochemical side: how the bioentities transform mechanical signals to chemical signals? The research field addressing the former question is usually overlapping with research in the field of biomechanics, and the latter with molecular biology.

In this mini‐review, I will focus on the former question and elaborate the fundamental concepts that hold across scales based on soft matter physics. In the following sections, I will start from intuitive phenomenology of material states and the force–motion relations, to concepts such as energy and conservation laws with a higher level of abstractness. To illustrate how these concepts perform in biology, I introduce up‐to‐date applications of several representative soft matter physics models. This review is aimed to give a unified landscape that could promote the synergy of biologists and physicists in the mechanobiology field.

2. BASIC CONCEPTS OF SOFT MATTER PHYSICS

2.1. Material state

Bioentities from tissues to proteins are all composed of soft matter. Soft matter refers to those materials consisting of self‐assembled mesoscopic units much larger than the atom size (Doi, 2013). In contrast to solid condensed matter, which is highly subjected to quantum effects because they are composed of atoms, structural changes of soft matter are dominated by thermal effects. Typical examples include polymers, colloids, liquid crystals, surfactants, etc. These large components interact with each other in specific ways, which give rise to the “soft” features on the macroscopic level. The interactions between each pair of units could last on some timescale τ due to physical constraints or vanish because of the motion under forces. The mechanism of the turnover of interactive bonds among pairs will render the unit motions reversible under forces applied on a timescale smaller than τ, but nonreversible otherwise. This former state is often intuitively referred to as “solid” and the latter one as “fluid” in some literature. However, these terminologies might mislead people to think that “solid” is something like a stone and “fluid” is something like water. In fact, in the rigorous definition of physics, solid refers to the materials with ordered microstructures such as crystals, whereas fluid refers to the materials with amorphous microstructures and includes both gases and liquids. For soft matter, fluids could also be rigid like glass once they are quenched below a critical temperature; and a “solid” state of a soft matter could also be amorphous in the microstructures, thus it is not truly a solid. A proper understanding of the material state of soft matter is essential to understanding mechanobiology and its complexity. Here, we introduce two ways to interpret the duality in the material states of soft matter.

One is from the perspective of intrinsic thermodynamics. Thermodynamics induces the spontaneous fluctuations of units with heat energy. If we track the position of each unit over time, their motion is confined by the interactive bonds in the vicinity of its original place at a smaller timescale, but they can diffuse as freely as Brownian particles (like molecules in water) at a larger timescale. The average distance the unit can move under thermodynamics increases with time in different fashions for the “solid” state and the “fluid” state. The particle trajectories in solids are caged in their local space permanently, but those in liquids move away from the origin and the distance increases with time (Figure 1a).

FIGURE 1.

Material states of soft matter. (a) Particle motions in a solid state and a fluid state. The displacement of particles on average over time is distinctive in solids and fluids. In a solid state, particles move in a limited range, while in a fluid state, particles move randomly away. (b) Elastic deformation under forces involves no dislocation of interparticle bonds, whereas viscous deformation under forces will change the conformation structure by breaking original bonds and establish new bonds. Elastic deformation is reversible once the force is gone and energy is conserved, whereas viscous deformation is irreversible, and energy is dissipated during the process. (c) Macroscopic phenomena for the material states in terms of three aspects. Relaxation: force relaxes under constant deformation. Creep: deformation changes under a constant force. Hysteresis: force–deformation relationships are different under the processes of loading, which is the increase in the force linearly with time (straight curves) and unloading (dashed curves), which is the decrease in the force linearly with time. (d) Two simple models for viscoelastic materials. Top: Maxwell model for viscoelastic fluids. Bottom: Kelvin–Voigt model for viscoelastic solids.

Another perspective is to observe the motion of units under applied forces (Figure 1b). In the “solid” state, since the interactive bonds last, the motion of units subjected to forces is reversible after the force is gone, and the energy is conserved before and after the force application. On the contrary, if the force lasts more than the characteristic timescale, some interactive bonds may break, and the units dislocate elsewhere and never return to the original positions once the force is gone. In addition, some energy is transferred to heat during the breakage of bonds and dislocation of units. Materials with reversible motions of units under forces are called “elastic,” while those with irreversible ones are “viscous.” The final irreversible state of the material is also called “plastic.” Since the elastic part has reversible deformations, the energy is conserved during the process, whereas the viscous part has irreversible deformations, and thus, energy is dissipated. Elastic behavior belongs to solids, and viscous behavior belongs to liquids.

The ability of a material to resist elastic deformation, that is, elasticity, could either be low or high. Materials with low elasticity are described as “compliant” as compared with those with high elasticity described as “stiff.” Likewise, the ability of a material to resist viscous deformation, that is, viscosity, could also be low or high. Materials with low viscosity are described as “soft” and those of high viscosity as “rigid.” For simple solids or liquids such as stone or water, they are either elastic or viscous and stable with the observation timescale under fixed pressure and temperature; however, soft matter materials are both elastic and viscous and their behaviors change with the timescale of observation. By the complexity in the microstructures of some soft matter, especially those composed of biocomponents, the timescale on which both elastic and viscous types of deformation coexist could be large. Hence, the soft matter involved in biology is mostly “viscoelastic.”

On a macroscopic scale, the material state of a soft matter could be probed from three aspects, namely relaxation, creep, and hysteresis. Relaxation refers to the force evolution with time under constant extent of deformation. Creep refers to the deformation evolution with time under a constant force. Hysteresis refers to the difference of force‐motion relationship between the loading and unloading processes.

Figure 1c shows the typical relaxation, creep, and hysteresis in elastic, viscous, and viscoelastic materials. Two typical viscoelastic materials are presented, namely viscoelastic fluid and viscoelastic solid. Viscoelastic fluids are fluids on the longer timescale but exhibiting elastic deformation on the smaller timescale, for example, asphalt mixtures and Silly Putty. Viscoelastic solids are solids on the longer timescale with additional energy dissipation during elastic deformation, for example, agarose gel and viscoelastic rubber. The distinction between these two types of viscoelastic materials could be discerned from their material structures as depicted in Figure 1d. Viscoelastic fluids could be ideally modeled by a joint of an elastic spring and a viscous damper in series (top, Maxwell model); conversely, viscoelastic solids could be modeled as a joint of the spring and the damper in parallel (bottom, Kelvin–Voigt model). In reality, viscoelastic materials could be modeled as combinations of series and parallel configurations of the springs and dampers, which are more complex than the two ideal models presented here.

2.2. Constitutive relations

As mentioned above, the motion of units in the material is related to the applied forces. The equations relating the local unit displacements to local forces are called constitutive relations and each type of materials is uniquely described by a constitutive relation. As a simple instance, the extension of an elastic spring is proportional to the applied force that stretches it, as depicted by the Hookean's Law in one dimension.

In soft matter, the motion of the units in response to forces could be decomposed to a reversible part and an irreversible part. In a simple material, the reversible part by can be characterized by the change of local deformation proportional to the forces as

$$F=E\text{∆}X,$$ (1)

where E describes the elastic stiffness, and ∆X, called strain, is the local change in distance between units relative to the original distance. As strain is the ratio of distances, it is a dimensionless quantity.

The irreversible part, however, can be described as

$$F=\mathit{\eta }△\dot{X},$$ (2)

where η is the viscosity and $△\dot{X}$ is the time derivative of strain, called strain rate, therefore in the unit of s ⁻¹.

According to Newton's third law on force and its reactive force, the force exerting to one component in the material must be balanced by another force with the same magnitude but an opposite sign. Therefore, the local forces appear in pair, called force dipole (Figure 2a) and they can be translated into a force pair per unit area with a direction, called stress (Figure 2b). In high dimensions, both stress and strain ∆X are tensorial quantities expressed as matrices (stress denoted as σ, Figure 2c). To notice, pressure is the isotropic part of the stress tensor and it is a scalar quantity without a direction (e.g., in two dimensions, pressure = Iso(σ) = (σ _xx  + σ _yy )/2). The elasticity and viscosity then are matrices of linear coefficients relating the strain ∆X (conventionally expressed as ϵ) or strain rate $△\dot{X}$ (conventionally expressed as $\dot{ϵ}$) to stress. Particularly, elasticity can be classified into the one resisting bulk deformation which changes the volume of an object, and the one resisting shear deformation, which does not change the volume. When bioentities are regarded as “soft,” one usually refers to their shear deformability.

FIGURE 2.

Force dipoles and stress components in the deformed material. (a) When the material is extended under an external force, every local part of the material is balanced by a pair of forces, with opposite signs; this pair is a force dipole. (b) For each local part of the material, stresses are defined as the pairs of force dipole exerted along each direction. A 2D view for a local component. Left: two normal stresses σ _xx and σ _yy along two orthogonal axes x and y. Right: two shear stresses σ _yx and σ _xy . Stress components can also have signs indicating their direction of deformation, for example, positive for contraction, negative for extension, as depicted in the figure. (c) Stress components can be represented in a tensor with their corresponding strain components. Elasticity and viscosity of the material are determined by how the stress tensor is related with the strain tensor and the strain rate tensor.

The theoretical models comprising only linear transformations from strains and strain rates are referred to as linear viscoelasticity theory. Linearity is rare in biological materials because of the complexity in their constituents and configurations, which are inhomogeneous and active. However, when stress, strain, or strain rates are small, the linear viscoelasticity can hold approximately.

The constitutive relations that map forces to spatial information of the material are crucial to mechanobiology. On one hand, measuring the spatial information of material becomes increasingly feasible owing to the enhancement of high‐resolution imaging techniques. Assuming some microstructure model of the material, one is able to establish a constitutive relation and infer the force profile from the spatial information profile, such as the position, velocity, geometry, and orientation of the elements that build up the materials (Nier et al., 2016; Roca‐Cusachs et al., 2017; Style et al., 2014). On the other hand, the active responses of the biological entities manifest themselves in changing microscopic conformations by synthesis or degradation of the components or remodeling their microstructures, which could be captured by spatial information profiles. Some active changes of bioentities such as dynamics of molecular motors (Veigel & Schmidt, 2011), folding/unfolding of proteins (Petrosyan et al., 2021), sol–gel transition of nuclear chromatins (Eshghi et al., 2021; Khanna et al., 2019), opening the ion channels on the nuclear envelopes (Goelzer et al., 2021; Matsuda & Mofrad, 2022), the alignment of filament bundles in the cytoskeleton (Scheff et al., 2021), or the secretion of components like collagens to extracellular matrix (ECM) (Chaudhuri et al., 2020; Humphrey et al., 2014; Vogel, 2018) have triggered a boom of interest in the mechanobiology field.

2.3. Rheology: Measuring mechanical properties

Forces inside the bioentities could be as small as piconewtons, and thus the strains could be small enough to hold linear viscoelasticity. Measuring the linear coefficients in the constitutive relations is a standard technical protocol in the field of rheology. Rheology is a phenomenological study of material deformation and flow under forces. Conventional rheology on the macroscopic scale has long been applied to probe the viscoelasticity of tissues and ECM (Forgacs et al., 1998; Janmey et al., 2007a; Petridou & Heisenberg, 2019). Meanwhile, its counterpart considering the probing into small objects on the intracellular scale has also been developed (Mason, 2000; Mason & Weitz, 1995). There are passive and active protocols designed for different purposes (Brau et al., 2007; Furst & Squires, 2017; Mao et al., 2022; Wirtz et al., 2009; Zia, 2018). The passive (micro)rheology utilizes the intrinsic fluctuations of particle positions embedded in the material, and one measures the mean square displacement of the particles. The displacement of an embedded particle between two time points increase differently for different microstructures of the materials. Assuming the material is in the equilibrium state (in a steady bath of thermal noises), one can relate the increase in the displacement with time to the material's elasticity and viscosity based on generalized Stokes–Einstein relations (Mason, 2000). Passive (micro)rheology has the merit of noninvasiveness, but it requires the material components to have spontaneous motions; therefore, the range of materials accessible by passive (micro)rheology is limited to those “soft” ones.

Alternatively, the elasticity and viscosity of the material could be measured by active (micro)rheology (Zia, 2018), in which external forces (usually magnetic, optic, or atomic) are applied to the material to result in oscillatory strains. Let us suppose the external stress has a time dependency as σ(t) = σ ₀sin(ωt). A pure elastic material with elastic modulus E should have the strain in the same oscillatory phase with the applied stress as ϵ _ela(t) = σ ₀sin(ωt)/E, whereas a pure viscous material with viscosity η should have the strain as ϵ _vis(t) = σ ₀cos(ωt)/η = σ ₀sin(ωt + π/2)/η, shifted in phase by π/2 from the stress. For a linear viscoelastic material, $ϵ\left(t\right)={ϵ}_0\sin \left(\omega t+\delta \right)$, and the stress–strain phase shift δ is between 0 to π/2, and this degree of phase shift depends on the frequency of oscillation ω. From the dependency of stress–strain profile on frequency, one can reconstruct the storage modulus (which describes elasticity) as

$$G^{\prime }=\frac{{\sigma }_0}{{ϵ}_0}\cos \delta$$

and loss modulus (which describes viscosity) as

$$G^{\prime \prime }=\frac{{\sigma }_0}{{ϵ}_0}\sin \delta$$

for different timescales (Meyers & Chawla, 2008), as frequency is in a reciprocal relation to time. Viscoelastic materials with different structures exhibit different dependencies of moduli on the timescales; therefore, this technique has been applied to identify distinct types of viscoelastic material. Unlike the passive microrheology, which requires the system to be in equilibrium for the usage of Stokes–Einstein relations, active microrheology can work for the force–motion relations in nonequilibrium systems (see a comparison between two methods in cytoskeletal rheology (Mizuno et al., 2007)), thus greatly enhancing the range of application in the field of biology.

Although passive and active (micro)rheology protocols have been extensively used in measuring the viscoelasticity of biological systems from tissues to nuclear condensates, their validity stands on the ground of linear viscoelasticity. Nevertheless, many biological matters show nonlinear viscoelasticity. The origins of the nonlinearity of material viscoelasticity could be generally classified into three categories: (1) the nonlinearity lies in the microscopic components themselves due to the intrinsic asymmetry in the force–deformation relationship of the molecular conformations (Janmey et al., 2007b; Oakley et al., 1998; Storm et al., 2005; Wen et al., 2007); (2) the nonlinearity rises from the transition of cooperative modes (network) of components resulted from the geometrical change as the deformation occurs (Feng et al., 2015; Fischer‐Friedrich, 2018; Onck et al., 2005); and (3) some dynamic processes directly modify the network of components, such as transient cross‐linkers (Chen et al., 2021; Mulla et al., 2019) and the active molecular motors (Koenderink et al., 2009).

As an outstanding example, cytoskeleton is featured by all these three mechanisms (Mofrad, 2009). Its contributing elements, filaments, belong to the family of semi‐flexible polymers, which have an intrinsic nonlinear force–deformation feature due to the reduced thermal undulations under stretching (Storm et al., 2005). Meanwhile, they exhibit transitions from bending modes to stretching modes featuring alignment of these filaments while being sheared (Reymann et al., 2016; Scheff et al., 2021); also, the transient cross‐linkers therein cause slow stress relaxation in the cytoskeleton (Broedersz et al., 2010a), similar to that of a soft glass, a family of liquids that exhibit metastable disordered microstructures (Sollich et al., 1997).

The prevalence of nonlinear viscoelasticity in biological materials stimulates great interest in developing different techniques and protocols for studying nonlinear rheology (Broedersz et al., 2010b; Ewoldt et al., 2008; Squires & Brady, 2005). As one of the typical ways, one can measure the differential modulus (which is the ratio between the change in stress over the change in strain) as a function of applied external stresses and compare it with the moduli measured in a linear (micro)rheology (Broedersz et al., 2010b). This method has been widely used into probing the nonlinear viscoelasticity in the cytoskeleton (Kollmannsberger et al., 2011) and ECM (Licup et al., 2015).

A substantial body of studies have been accumulated on the rheology of biological systems at different scales, such as in embryonic tissues (Perepelyuk et al., 2016; Petridou & Heisenberg, 2019), ECM (Cramer et al., 2017; Massensini et al., 2015; Staunton et al., 2016), cytoskeleton (Mofrad, 2009; Rigato et al., 2017), nuclei (Celedon et al., 2011; Dahl et al., 2004; Guilak et al., 2000; Lherbette et al., 2017), and even scales below (Caragine et al., 2022; Michieletto & Marenda, 2022; Seelbinder et al., 2021; Zidovska, 2020), to mention a few. An important note here is that the protocols for probing the mechanical properties of materials could be rather diverse for biological systems, but the underlying principles remain the same as conventional rheology. One can either relate the spontaneous fluctuations in the component motions to the strength of noises (such as temperature) in a passive way or relate the intended motion to the applied forces in an active way.

2.4. Energy

Energy is an essential concept not only for soft matter physics but for the operation of the universe. As the microscopic components of the material move under forces, the forces do work to the material, which could be stored by the new configuration as elastic potential energy or dissipated through viscous deformation.

At equilibrium, the energy is minimized at the “best” conformation of the material microstructures. These “best” configurations are the ones with the highest stability. The resistance to deviation from the “best” configuration is the origin of elasticity and the resistance to the energy loss during configuration change is the essence of viscosity. Understanding the energy of a system can help predict the fate of the material under forces and various conditions.

In different systems, energy could be defined differently. For thermodynamics‐driven solids, the energy is usually the Gibbs free energy $\mathcal{G}$, which is the enthalpy $\mathcal{H}$ (related to heat) minus the temperature T times the entropy S (related to the probability of a configuration), that is, $\mathcal{G}=\mathcal{H}-TS$. The biopolymer solutions like cytoskeleton minimize the Gibbs free energy $\mathcal{G}$ G, by either lowering the enthalpy $\mathcal{H}$ H through mode‐changing deformation of the monomers or increasing the entropy S through rearrangement of the connections between monomers for a given temperature (Janmey et al., 2007b; Pujol et al., 2012). For liquid systems at a constant volume, the energy is defined to be the Helmholtz free energy $\mathcal{F}$ F, which is the internal energy $\mathcal{U}$ U (including energy for particle velocity and interaction among particles) minus the temperature T times entropy S, that is, $\mathcal{F}=\mathcal{U}-TS$. The intracellular organelles are formed by minimizing $\mathcal{F}$ F to reach the steady configuration that can separate different molecules to different compartments without membranes. Such concepts of phase separation (e.g., liquid–liquid phase separation and microphase separation) have been explored recently in the studies of intra‐ and subcellular organelles such as nuclei and chromosomes (Banani et al., 2017; Hyman et al., 2014; Rana et al., 2021; Ryu et al., 2021).

The utilization of energy concept could also be useful to explain the dynamics on a larger scale. For systems with higher complexity where thermal effects are insignificant to the dynamics, the minimized energy at the thermal equilibrium could be set as a reference value, leaving the energy variation to account only for macroscopic dynamics. A typical example is the 2D vertex type of models of confluent tissue mechanics for epithelial monolayers (Bi et al., 2015, 2016; Garcia et al., 2015; Kim et al., 2021). This model defines an energy $\mathcal{E}$ E as the elastic energy for the cell area A and perimeter P deviating from their equilibrium values: , where the summation ∑ is over all the cells in the system and coefficients k _a and k _p scale with the stiffness resisting the area change ∆A _i (bulk deformation) and the perimeter change ∆P _i (shear deformation). Given the defined elastic energy, the elastic force $F\left(x\right)=‐\partial \mathcal{E}/\partial x$, where ∂x is the change in positions of cell boundaries, drives the cells in the space. By minimizing this energy, simulated confluent tissues are confined to cell area and shapes of equilibrium values and present solid–liquid phase transitions independent of cell density (Bi et al., 2015; Merkel & Manning, 2018).

Besides the conventional elastic potential energy of bulk and shear deformations, other types of potential energy could also play important roles. For instance, nematic energy has drawn much attention in tissue and cytoskeleton studies recently (Balasubramaniam et al., 2022; Doostmohammadi et al., 2018; Saw et al., 2018). The nematic energy (also named as distortion free energy or Frank free energy) describes the elastic potential energy when the elongated components are deviated from a configuration of alignment. This tendency of components to align is the essential physics in the liquid crystals, a type of soft matter commonly encountered in daily life (Khoo, 2022). The existence of nematic energy brings about rich dynamics that regulate the alignment and polarity of components, which has been claimed to play a role in tissue homeostasis (Guillamat et al., 2022; Saw et al., 2017, 2018).

2.5. Dynamics

Despite its central role of energy in the operation of our universe, it is not easy to measure and formalize in reality, especially for complex systems. Instead, dynamics could be amenable to observe and found with phenomenological descriptions. Principally, dynamics (change with time) can be captured from three aspects:

• Mass conservation: The change of density ρ of components with time in a local space should be balanced by the generation the components (negative if degeneration), or the flux of the components into the space (negative if flux out):

  $$\frac{\mathit{d\rho }}{\mathit{dt}}=\text{flux in}+\text{generation.}$$ (3)

  This partial differential equation form is also called the continuity equation in continuum mechanics.
• Momentum conservation (force balance), the balance between the active forces F _a and the inertial force (in relation to the acceleration of the component), the frictional force (in relation to the velocity of the component), the potential force (in relation to the stored energy in the systems), noises (such as thermal fluctuations):

  $$\underset{\text{inertia}}{\underbrace{m\frac{d^{2}x}{d{t}^2}}}+\underset{\text{friction}}{\underbrace{\eta \frac{\mathit{dx}}{\mathit{dt}}}}+\underset{\text{potential}}{\underbrace{\frac{\partial \mathcal{E}}{\partial x}}}+\underset{\text{noise}}{\underbrace{\xi \left(t\right)}}=-F_{\mathrm{a}}.$$ (4)

  In biological systems, the inertia term (where m refers to the mass of the particle) in the force balance equation is always neglected because the mass of biological entities is too small to give an inertial force comparable to other force terms. One could notice that the coefficient η in the friction term has the same dimension with the viscosity in Equation (2); therefore, the friction term includes the impact from viscous deformation. The potential term could have a wide range of candidates, including but not limited to elastic energies.

• Energy conservation: During the dynamics, total energy of the system and the environment remains constant. There undergoes the exchange of energy between different forms. A textbook example could be the photosynthesis process in which the photon energy is transformed to chemical energy. A more relevant example in mechanobiology is the transformation of the biomechanical energy carried by myosin motor proteins to the elastic energy stored in the contraction of filaments in eukaryotes (Sweeney & Houdusse, 2010).

3. APPLICATION TO BIOLOGICAL SYSTEMS

The notions and principles in the last section by no means touch on the subtleties of real biology. Biological systems of particular interest are only a part of the total system where these conservation laws hold. The application of these physics‐based concepts to biology requires careful considerations depending on systems and purposes.

3.1. Choice of assumptions

The first challenge lies in the proper choice of assumptions about whether, and to what extent, the conservation laws could be relaxed. Mass conservation is robust at the scale of mechanobiology where no transformation between energy and mass occurs; a typical violation of mass conservation is the nuclear reaction, which is irrelevant to our interest. Energy conservation is so far considered trivial in the active systems because the bioentities could either absorb energy from the environment or release “active energy” from the chemical energy with a myriad of mechanisms.

Momentum conservation is the most heavily considered part in mechanobiology‐relevant problems. As explained above, inertia is always considered as small as compared with other forces for the rigorous reason that this biomaterial has a low Reynolds number. The Reynolds number is a dimensionless quantity describing the strength of inertia relative to the viscosity. A low Reynolds number (far smaller than 1) means that laminar flow without turbulence dominates the dynamics. Soft matter are characteristic of a low Reynolds number, so the inertial effect is usually neglected as a convention. This lack of inertia in the dynamics is also called “overdamped.”

Another important assumption lies in whether or not the flow of media caused by the components affects the dynamics of the components (Figure 3, see reviews in Banerjee & Marchetti, 2019; Jülicher et al., 2018; Marchetti et al., 2013; Nejad et al., 2021; Shaebani et al., 2020). This is relevant especially for system components with nonelastic motions. For example, when the birds fly in the air, they are subjected to frictional forces from the air, but the impact of air flow caused by the birds is negligible on the large length scale of the bird size. If the flow of media is negligible in the dynamics of the components, then it is called a “dry” model. Dry dynamics breaks the law of momentum conservation because the momentum transferred from the components (bird) to the media (air) is dropped. The diffusion phenomena observed in daily life is a typical instance of dry dynamics. Called Brownian dynamics, it is an overdamped version of Langevin dynamics, which describes particle motions under friction in a dilute media with Gaussian noises either from thermal or active sources. Brownian dynamics is not only applied to the molecular dynamics in a dilute solution, but also to the other systems such as flowing confluent tissues where the substrate embedding the cells is assumed to resist their motions by friction but does not flow with the cells (Bi et al., 2015, 2016). In the past decades, dry dynamics have been extensively explored for their rich macroscopic phenomena, despite the simplicity in the model implementation (Chat́e et al., 2008; Ginelli, 2016; Toner & Tu, 1995; Vicsek et al., 1995). The subjects cover collective behaviors of microtubules (Sanchez et al., 2012; Sumino et al., 2012), bacterial swarming (Ben‐Jacob et al., 1995; Be'er et al., 2020; Chen et al., 2017; Ginelli et al., 2010), solid‐to‐fluid transitions of confluent tissues (Bi et al., 2015, 2016; Chiang & Marenduzzo, 2016; Loewe et al., 2020; Merkel & Manning, 2018), collective cell migration (Alert & Trepat, 2020; Banerjee & Marchetti, 2019; Belmonte et al., 2008; Camley & Rappel, 2017), and herding of animals such as birds and sheep (Giardina, 2008; Ginelli et al., 2015; Toner et al., 2005; Vicsek & Zafeiris, 2012).

FIGURE 3.

Illustration of dry and wet dynamics. In dry dynamics, the media does not flow with the particles and thus momentum is not conserved in the model.

In contrast, it is assumed in the “wet” dynamics model that the flow of media is significant to the dynamics of components as if they deform together as a whole. The terminology “hydrodynamic effect” often refers to this impact from media flow to the dynamics of components. Then a model with momentum conservation is required. Newtonian fluids have linear viscosity following both mass and momentum conservation, and therefore, they can be described by the complete Navier‐Stokes equations. For the overdamped case (Reynolds number close to zero), they are reduced to the Stokes equations. Coupling “active” components to Stokes types of models at different levels of coarse graining has become a rising research interest in the past decades. A series of new theoretical methods have been formalized. To mention a few, microscopic “squirmer” models explain the microorganism swimming (Cisneros et al., 2010; De Corato et al., 2015; Downton & Stark, 2009), “active gel” theory (Joanny & Prost, 2009; Kruse et al., 2004; Prost et al., 2015) explains the continuum mechanics of cytoskeleton, and “active fluid” or “active suspension” models (Dunkel et al., 2013a, 2013b; Slomka & Dunkel, 2015) describe the long‐range flow in multicellular organizations.

Choosing the type of dry or wet dynamics for modeling depends on the timescale of interest specific to the system. On a large enough timescale, the hydrodynamic effect could always be neglected. On a smaller timescale, both types of models could reveal nontrivial information. As a typical example, bacterial swarming (Copeland & Weibel, 2009; Lauga, 2016) could be either described by the dry dynamics of the self‐propelled rods (Ginelli et al., 2010; Meacock et al., 2021; Peruani et al., 2006; Wensink et al., 2012), or wet dynamics of active nematic materials (Chen et al., 2017; Dunkel et al., 2013b; Hu et al., 2015). Dry models describe the huge density fluctuation in space and active turbulence (without inertia term) (Wensink et al., 2012). In a comparison between dry and wet dynamics (Chen et al., 2017), the authors found that the dry part induces collective oscillation in the alignment of bacteria, but the wet part generates elliptic trajectories of the collective oscillation.

3.2. Other subtle problems

The force balance in Equation (4) is the force balance for a mass point. However, in a material, forces are transmitted through components at different locations. This brings complexities in applying existing physics models to the system of biological interest.

Firstly, a material occupies a space, which is defined by the boundaries. Even for simple elastic materials such as matured ECM, applying different boundary conditions to the system induces qualitative change in the biomechanical responses. For example, deformation of elastic substrates caused by cell traction forces has a critical thickness, less than which cells only feel the hard supportive base below the substrate and behave as if on a stiff substrate (Buxboim et al., 2010; Maloney et al., 2008; Merkel et al., 2007; Sen et al., 2009); if the substrate has frictional slipping at the boundary of the supportive base, there emerges slow force propagation which mediates cell divisions (Kawaue et al., 2023; Lou et al., 2022). Another boundary issue is the curvature of boundary. Force balance in curved materials such as epithelial tissue layers involves additional anisotropic stresses that lead to irregular deformations of cells (Harmand et al., 2021; Lou et al., 2023; Messal et al., 2019), increases tissue rigidity (Sussman, 2020), and affects the nematic flow patterns (Bell et al., 2022; Connon & Gouveia, 2021).

Secondly, the breaking of spatial symmetry such as inhomogeneity and anisotropy significantly influences force profiles. As mentioned above, curvature of boundary induces anisotropy in surface stress; besides, various types of anisotropy widely exist in biological systems. The most prevalent anisotropy across the scales is shape anisotropy, polarity (velocity anisotropy), and chirality or “handness” (Tee et al., 2015; Utsunomiya et al., 2019). The foundation of the newly booming field “active nematics” (Balasubramaniam et al., 2022; Doostmohammadi et al., 2018; Sazer & Schiessel, 2018) lies in the stress anisotropy determined by the orientation of the components, and this stress anisotropy could generate rich and biologically relevant consequences such as the remodeling and stiffening of matrix and cytoskeleton (Feng et al., 2015; Reymann et al., 2016; Scheff et al., 2021), the dynamics of topological defects that induce cell death and differentiation (Guillamat et al., 2022; Saw et al., 2017), pressure inhomogeneity around the topological defect that affects cell division rates (Delarue et al., 2014; Dolega et al., 2017), etc.

The key to bridging physics to biology is the balance of simplicity and complexity. Physicists seeking simple explanations start from a solid knowledge of physics and upgrade their understanding by layering up controllable ingredients. Biologists interested in complex life phenomena start from mutually consistent empirical evidence and upgrade their understanding by converging scattered pieces to a cohesive cause–effect story. The common goal of both groups is to find the nontrivial biological ingredients that generate rich phenomena from simple physics. Shape anisotropy and alignment of components are such typical ingredients with which people upgrade the liquid crystal theory to the biological systems, by adding the active stresses along the anisotropic orientations.

3.3. Summary

In this mini‐review, I have introduced the essential concepts of soft matter physics involved in mechanobiology across scales and mainly focused on the force–motion principles underlying soft materials. Several popular branches of physics models seasoned with “activeness” have been highlighted. The works mentioned here are far from covering the whole scope of the application of soft physics models to mechanobiology. Some sophisticated concepts such as viscoplasticity (Nam et al., 2016) and yield (Dyson et al., 2012; Hopkins et al., 2022), which are often seen in biomaterials, have not been included. In conclusion, the success of applying physics theory to the mechanobiology problem lies in the proper choice of model assumptions, careful treatment on boundaries, and pinpointing the biologically relevant ingredients that bring about rich phenomena in mechanics and give insights into experiments.

Lou, Y. (2023). Appetizer on soft matter physics concepts in mechanobiology. Development, Growth & Differentiation, 65(5), 234–244. 10.1111/dgd.12853