Statistical mechanics of cell aggregates: explaining the phase transition and paradoxical piezoelectric behavior of soft biological tissues

Abstract

Piezoelectricity in biological soft tissues is a controversial issue with differing opinions. While there is compelling experimental evidence to suggest a piezoelectric-like response in tissues such as the aortic wall (among others), there are equally compelling experiments that argue against this notion. In addition, the lack of a polar structure in the underlying components of most soft biological tissues supports the latter. In this paper, we address the collective behavior of cells within a two-dimensional cell aggregate from the viewpoint of statistical mechanics. Our starting point is the simplest form of energy for cell behavior that only includes known observable facts e.g., the electrical Maxwell stress or electrostriction, resting potential across cell membranes, elasticity, and we explicitly exclude any possibility of electromechanical coupling reminiscent of piezoelectricity at the cellular level. We coarse-grain our cellular aggregate to obtain its emergent mechanical, physical, and electromechanical properties. Our findings indicate that the fluctuation of cellular strain (E) plays a similar role as the absolute temperature in a conventional atomistic-level statistical model. The coarse-grained effective free energy reveals several intriguing features of the collective behavior of cell aggregates, such as solid–fluid phase transitions and a distinct piezoelectric-like coupling, even though it is completely absent at the microscale. Closed-form formulas are obtained for key electromechanical properties, including stiffness, effective resting potential, critical E²-temperature (or fluctuation) for solid–fluid phase transitions, and apparent piezoelectric coupling in terms of fluctuation and electric potential regulated by active cellular processes.

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1. Introduction

Cell aggregates or tissues are composed of individual cells, intricately organized with notable variability in their shapes, sizes, and spatial arrangements. Each constituent cell is structurally complex, featuring a bilipid membrane interspersed with ion channels and molecular pumps. Enclosed within this membrane are cytoskeleton, cytoplasm, and various organelles. The hierarchical structures and complex interplay of biophysical processes within cells give rise to intriguing electromechanical properties of single cells and cell aggregates. In particular, the bilipid membrane and cytoskeleton provide shear resistance and maintain the structural integrity of a cell;¹ the combined activities of numerous ion channels and molecular pumps induce a long-term, steady-state electric potential difference across the cell membrane.² The propagation of changes in transmembrane potentials mediates cellular signaling and regulates critical biophysical processes.³ Conversely, mechanical deformations can regulate ion channels and pumps, inducing cellular bioelectricity within cell aggregates. These electromechanical couplings provide the fundamental physical mechanism underlying biological functions, including cellular adhesion, migration, differentiation, and mechanosensing.^4–9

Cells in an aggregate interact with each other both mechanically and electrically. The resulting electromechanical properties of cell aggregates or tissues are derived from the collective behavior of individual cells, leading to properties and functionalities that are significantly richer than those of a single cell. For example, cell aggregates reversibly transition between a rigid solid-like phase and a floppy fluid-like phase referred to as a solid–fluid phase transition, which plays a critical role in the biological function of living tissues.^10–17 This phase transition can be actively regulated by transmembrane potential.^18–21

Currently, several mechanisms have been proposed to explain the solid–fluid phase transition in cell aggregates. These include a topological transition which promotes tissue fluidization by relaxing elastic stresses over longer time scales;^10,22,23 the sliding of coherent cells within the aggregate, where external forces induce gradual rearrangements of cell conformations;²¹ and transitions driven by the density of multicellular vertices (rosettes) and intracellular tensions.¹¹ Within a tissue, cellular deformation and electric potential exhibit considerable fluctuation and variability. The fluctuation in cellular shape and size can be readily quantified, providing valuable insight into the physiological stage of the tissue.²⁴

The true nature of the electromechanical behavior of soft biological tissues has attracted repeated controversy. Claims have been made regarding the piezoelectricity of these tissues cf. Fukada and reference therein²⁵ and even the presence of ferroelectricity.²⁶ Convincing counter-arguments have been made against this by Lenz et al.,²⁷ who argue that the observable electromechanical coupling is due to Maxwell stress or electrostriction. In an interesting turn of events, in a more recent work, we turn full circle where Ikushima et al.²⁸ highlight ultrasound experiments that do appear to indicate the presence of piezoelectricity in aortic wall tissue.

Existing approaches to modeling cells and their aggregates can be broadly divided into three categories: agent-based models, continuum models, and statistical models. The agent-based approach treats a single cell as a homogeneous, isotropic, elastic, and spherical body.^29–32 While this approach is valuable for studying the interaction of individual cells with each other and with their environment,^33–35 its applicability to the macroscopic behavior of cell aggregates remains limited. Continuum models for cell aggregates,^36–41 by contrast, are well-suited for modeling large-scale phenomena where aggregate properties vary smoothly over length scales spanning several cell diameters. These models effectively capture the macroscopic response of cell aggregates and can be extended to incorporate additional physical phenomena such as intercellular interactions and formation of cell aggregates.⁴² For instance, a continuum description of cell motility, driven by cell–cell and cell–ECM (extracellular matrix) interactions, was developed using a non-local interaction term to account for adhesion between cells and between cells and the ECM.^43–45 Despite these successes, continuum approaches fall short in capturing the inherent variability in cellular deformation and electric potential within tissues. In contrast, statistical models provide a robust framework for addressing these limitations.^46–49 By applying the framework of statistical mechanics to cell aggregates, we can effectively describe the mechanical and electrical fluctuations observed in tissues, bridging the gap between microscopic cellular properties and macroscopic tissue behavior. As discussed below, this approach also allows for a quantitative analysis of deformation and electric potential fluctuation, offering deeper insights into the emergent electromechanical properties of cell aggregates.

The method of statistical mechanics has been employed in a number of studies on cells and their aggregates. A recently developed theory^50,51 addressed the active processes in cells to provide insights into cell size distribution. They extended that work to include electrical fields to establish the threshold electrical fields that can be sensed by a cell.⁵² The work¹⁰ based on non-equilibrium statistical mechanics and a 2D vertex model was able to capture mechanical instabilities in cell aggregates. The study in ref. 53 for active matter (living cells) focused on steric and aligning interactions and interactions driven by shape changes. The impact of thermal fluctuations in polarization on the renormalization of the flexoelectric coupling coefficient of cell membranes has also been studied using statistical mechanics.⁵⁴ However, none of the existing models incorporate both the mechanical and electrical fluctuations and inter-cellular interactions for cell aggregates.

In this work, we propose an electromechanical model to describe both single cells and cell aggregates based on structural symmetries and derive its macroscopic continuum limit. Notably, our starting point for the microscopic model (at the cell level) does not include any piezoelectric behavior and simply takes cognizance of the fact that there is a resting potential across the cell and the presence of electrostriction (which is exhibited by all materials). Fluctuations in geometric parameters, such as cell area and aspect ratio, can be effectively quantified through the analysis of tissue micrographs. Notably, the fluctuation in strain—later referred to as the E²-temperature due to its analogous role to absolute temperature in classical thermal physics—serves as a quantitative measure of the inherent system variability.⁵⁵ Our resulting coarse-grained electromechanical model for cell aggregates is particularly well-suited for applications to epithelial tissues, which consist of flattened, tightly packed cell layers. However, our insights are broader and transferrable to other tissue configurations. Based on the coarse-grained model, we investigate the macroscopic electromechanical behavior of cell aggregates, demonstrating that effective electromechanical coupling and solid–fluid phase transitions arise from the collective cellular interactions. These resolve the existing paradox in the literature. The emergent properties depend on the E²-temperature. The quantitative dependence of effective resting potential, tissue stiffness, T1-transition rate, viscosity, and critical point for solid–fluid phase transition on the E²-temperature could be validated by experimental observations. Such comparisons open the door to understanding how fluctuation and variability contribute to tissue behavior in different biological contexts, making the E²-temperature an experimentally testable concept for linking microscopic fluctuations with macroscopic emergent properties.

1.1. Central premise

Following the paradigm of statistical mechanics, we treat a cell aggregate as a canonical ensemble of many cells. We borrow the terminologies and methodology from statistical mechanics with the caveat that the fluctuation or variability of the system mainly arise from biological active processes (e.g. growing, diffusion, etc.), defects, inhomogeneities, among others, instead of the actual thermal agitations. Therefore, the benchmark energy scale (k̃_BT = 1/β) used in this work is a phenomenological constant; the fluctuation or variability in cellular states arises not only from thermal agitations but also from many other passive and active processes that entail “equilibrium” of the system at the observation time-scale. The collection of cellular strains and electric potentials in a cell aggregate are assumed to describe the microstates of the system. For their collective properties in a cell aggregates, we first propose a single-cell Hamiltonian based on the underlying structural symmetries and experimental observations of their electromechanical responses. We employ the mean-field approximation, which allows for an analytical solution of the model and yields a closed-form expression for the effective free energy of cell aggregates. Based on this effective free energy, we present several experimentally testable predictions, including the macroscopic electromechanical properties, the solid–fluid phase transitions observed in cell aggregates, and their dependence on the E²-temperature and average electric potential regulated by active processes.

2. An electromechanical model for cell aggregates

We are interested in the macroscopic electromechanical response of a cell aggregate as illustrated in Fig. 1(a). We first introduce independent state variables for describing physical and geometrical configurations of cells and their aggregates. Fig. 1(a) shows a schematic of a typical amorphous cell aggregate. We idealize the cell aggregate as a tessellation of a plane by equal-area polygons as illustrated in Fig. 1(b). Denote by C_i⊂[Doublestruck R]² the polygon/domain occupied by the ith cell, A the area of the polygon C_i, and c_i the spatial positions of the centroids of cells. For simplicity, we assume that each cell precisely occupies the Voronoi cell associated with {c_i: i = 1, …, N}.⁵⁶ Though it is tempting to employ the centroids of cells as the fundamental state variables, a reasonable physical model or Hamiltonian for the tessellation in terms of {c_i: i = 1, …, N} must be interacting and permutation invariant, and hence not amenable to systematic perturbation methods. On the other hand, for cell aggregates illustrated in Fig. 1(a), we see that the shapes of cells and statistical properties of aspect ratios are good indicator of different physiological stages of cell aggregates (or tissues) which, presumably, dictate the physical properties of cell aggregates as well.^55,57,58 Therefore, we choose the deformation or strain as one of the independent state variables for the cell aggregate.

Fig. 1 Cell aggregate and single cell. (a) A typical cell aggregate (plated Madin-Darby Canine Kidney cells, Courtesy of K. Irvine). (b) Geometrically, a cell aggregate is regarded as a tessellation of a plane by polygons. (c) Schematic of a single cell occupying a regular hexagonal domain (n = 6) and (d) a deformed hexagon, where c, x_j, y_j is the position of the centroid of the cell, the jth vertex of the cell in undeformed and deformed configuration, respectively.

Unlike crystalline solids for which there exists a natural reference configuration (i.e., crystalline lattice), at least locally, for defining the deformation and strain, a cell aggregate is amorphous whose deformation or strain, strictly speaking, cannot be defined for a single current configuration. Nevertheless, neglecting the compatibility requirement for neighboring cells, “strain” could be defined for each individual cell by a few alternative procedures. The essence of these procedures is such that the introduced strain tensor E ∈ [Doublestruck R]^2×2_sym is a measure of orientation and aspect ratio of individual cells and the underlying arrangement of cells. In particular, we require that the strain tensor E ∈ [Doublestruck R]^2×2_sym satisfy the following.

(R1) The strain E = 0 if the cell is a regular polygon.

(R2) For small strains with |E| ≪ 1, the ratio of eigenvalues of strain tensor E ∈ [Doublestruck R]^2×2_sym measures the “aspect ratio” of the cell.

(R3) For small strains with |E| ≪ 1, the eigenvectors of strain tensor E ∈ [Doublestruck R]^2×2_sym defines the “orientation” of the cell.

As a tessellation of a plane, cells in aggregates occupy polygonal domain with n vertices y_j (j = 1,…,n). As illustrated in Fig. 1(a), typically n = 4, 5, 6, 7 from experiments. Suppose a cell occupies an irregular hexagon as in Fig. 1(d). We choose the regular hexagon of the same area in Fig. 1(c) as the reference undeformed configuration.

Below we introduce two procedures to identify the strain tensor E associated with a single cell in the regime of small strain (|E| ≪ 1).

(i) Let x_j (j = 1,…,n) be vertices of the regular n-gon centered at the origin:

where is such that the area of the regular n-gon equals the area of the cell. By the method of least square, we can in general define a deformation gradient F ∈ [Doublestruck R]^2×2 and a translational vector u₀ ∈ [Doublestruck R]² for the cell:
Let u_j = y_i − x_i be the displacement of the ith vertex and H = F − I the unsymmetrized strain (I ∈ [Doublestruck R]^2×2 is the identity matrix). We write L(F, u₀) as

By the first-order necessary conditions we have

which imply that

From classical continuum mechanics, the linearized symmetric strain for the cell can be written as

(ii) A second procedure to determine the strain tensor is through the normalized second moment of the current position vector. For a polygon illustrated in Fig. 1(d), without loss of generality we assume the centroid is at the origin: Then the normalized second moment tensor is given bywhere is the second moment tensor of a regular n-gon of the same area (cf. Fig. 1(c)). Suppose that the current shape is obtained by deforming the regular n-gon with vertices given by x_j in (1), i.e., y_j = Fx_j. The second normalized moment tensor is given by

Then the strain tensor may be defined as

E = (B)^1/2 − I, (4)

which clearly fulfills the previously mentioned requirements (R1)–(R3).

We remark that strains defined by (2) and (4) coincide to the leading order for small strains with |E| ≪ 1 and both of them have been used to quantify experimental and numerical results.^55,57 Also, for hexagons the shape index⁵⁷ or perimeter (p)-to-sqrt of area ratio can be expressed as a function of strain E:

2.1. Strain from rearrangement of cells

As for crystalline solids, significant strain can be generated by structural rearrangement of cells from the viewpoint of centroid positions. Such a transformation strain cannot be characterized by the two definitions ((2) and (4)) of strain tensor. As illustrated in Fig. 2, in a rearrangement process called T1-transition commonly observed in the development of epithelial tissues, neighboring cells exchange their positions through a reorganization of intercellular junctions without significant changes in the shape of individual cells. Though the final configuration in Fig. 2(c) in T1-transition is a deformed configuration relative to the initial configuration in Fig. 2(a), the strain calculated by either (2) or (4) vanishes for every cell. To remedy this issue, we observe that, as a cluster of cells, the final configuration in Fig. 2(c) can be obtained by compressing (and stretching) the initial configuration in Fig. 2(a) in y-direction (in x-direction). The strain associated with this deformation process may be identified aswhere the T1-transition strain γ* ∼ 1 is beyond the regime of small strains and will be determined later from empirical data.

Fig. 2 T1-transition in cell aggregates. Cell aggregates undergo rearrangement in which neighboring cells exchange their positions through a reorganization of intercellular junctions without significant changes in the shape of individual cells and transition from (a) initial configuration to (b) intermediate configuration to (c) final configuration.

To summarize, we note that both definitions of the strain tensor, (2) and (4), encounter difficulty in consistently capturing the local structural rearrangements of cells. This limitation arises from the inherent complexity of cellular configurations. In this work, we take a pragmatic approach and treat the strain tensor E as an “order parameter” as in the framework of Landau's phenomenological theory.^59,60 In this context, the precise mathematical definition of strain tensor E beyond the regime of small strain is secondary, as long as it qualitatively reflects the key geometric characteristics—namely, the shape changes and structural arrangement of the underlying cells. By adopting this perspective, the strain tensor serves as a convenient and effective state variable for quantifying the geometry and arrangement of cells in a cell aggregate and can be confidently computed by either (2) or (4) in the regime of small strain.

In addition to elasticity, we will consider electric interactions between cells. Cell aggregates, in spite of adhesion between them, in general, are immersed in an extracellular matrix (ECM) whose electric potential may be assigned as the ground potential. Ionic channels, i.e., specialized protein structures embedded in the cell membrane, allow selective passage of ions (e.g., Na⁺, K⁺, Ca²⁺, Cl⁻, etc.) across the membrane. This selective transport ensures and maintains a resting membrane potential ξ*, which is essential for numerous biological functions.² For simplicity, we assume the interior of each cell is of equipotential. The actual potential of the ith-cell, denoted by ξ_i, could be significantly different from the resting potential ξ* due to electromechanical coupling and other passive or active transport processes in cell aggregates.

In summary, for a phenomenological model of cell aggregates we assume that the state of the cell aggregates are completely described by the strain tensor and electric potential of cells:

where the constraint TrE = 0 arises from the incompressibility of cells. For brevity and future convenience, denote by s, s̄, s′ the collection of state variables of N-cells, the average, and their deviations:

2.2. Energetics

In line with standard practices in physical modeling, we formulate the electromechanical model for cell aggregates via an energy function. As illustrated in Fig. 1(d), we consider a single cell from the aggregate with strain-electric potential and propose that the cellular energy is given bywhere A is the area of the cell, and μ, γ* (∼1), ε, λ > 0 and ξ*, K are constant model parameters. Physically, the energy terms in (9) are motivated from the following consideration. The first term characterizes the strain energy; the specific expression is the minimal form that is isotropic, thermodynamically stable, and admits the regular hexagonal tessellation (cf. Fig. 2(a)) and its equivalent configuration after T1-transition (cf. Fig. 2(c)) as the strain-energy minimizing states. In particular, the constant μ may be recognized as the shear modulus (for small strain). The second term is the electric field energy with ξ* being the resting potential (at the absence of electromechanical coupling, i.e., K = 0), ε the dielectric constant, and λ a length scale comparable to the thickness of cell membrane. The electric field energy penalizes the deviation of interior cellular potential from the resting potential. The last term K(ξ − ξ*)|E|² is for capturing the coupling between electrical field and mechanical strain. We remark that the proposed single-cell energy (9) represents the minimal isotropic electromechanical model that achieves the following: (i) it effectively reproduces the observed T1-transitions in cell aggregates, featuring with the characteristic rearrangements during tissue remodeling, and (ii) it incorporates nontrivial electromechanical coupling, allowing for a meaningful interplay between mechanical deformations and electrical responses. This balance of simplicity and functionality makes the proposed model (9) a robust foundation for exploring the electromechanical properties of tissues.

We next consider cell aggregates that are nonuniformly deformed or charged. Let be the index set that contains all neighboring cells in contact with the ith cell. In account of nonuniformity, we postulate that the interaction energy for the ith cell is:

where the length scale l is comparable to the overall size of a cell, and constant J^elast > 0 (resp. J^elect > 0) measures the increase of energy when the strain (resp. electric potential) of a cell is different from that of neighboring cells. In the context of the Ising model, J^elast and J^elect are referred to as the exchange constants. Combining (9) with (10), we identify the Hamiltonian of an N-cell system aswhich serves as the foundation for our statistical mechanics model.

We will follow the standard procedure of statistical mechanics to analyze the cell aggregate. In the classical statistical and thermal physics, the critical concept of (absolute) temperature reflects the fluctuation of energy associated with microstates of the system, whereas entropy measures the number of accessible microstates of the system. In thermal equilibrium, the probability distribution over all admissible microstates of the system is dictated by the Second Law, which can then be used to determine macroscopic properties and thermodynamic relations of the system. To apply this approach to the electromechanical model (11) of a cell aggregate, we start from the following hypotheses.⁶¹

(H1) The variability or fluctuations in state variables (E, ξ) are macroscopically homogeneous within the system (a cell aggregate) and can still be characterized by a single benchmark energy scale k̃_BT = 1/β.

We remark that unlike the classical systems such as ideal gases, the stochasticity in cell aggregates has many contributions including but not limited to thermal agitations, passive and active biological processes (e.g., growing, diffusion, etc.), among others. Therefore, the benchmark energy scale k̃_BT = 1/β should be understood as a phenomenological parameter instead of the actual thermal energy scale that originates from the oscillations of constituent molecules or atoms. Nevertheless, for physiological relevance we can still interpret T as the absolute temperature of the system. Meanwhile, k̃_B should be regarded as an empirical constant in parallel to the classical Boltzmann's constant. Similar ideas have been used in the literature to describe novel phenomena in a range of physical systems, including granular media⁶² and plasticity in polycrystalline materials.⁶¹

(H2) In quasi-static processes, the cell aggregate may exchange energy with the environment and stay in instantaneous equilibrium with a constant and uniform temperature T, and hence follows the Boltzmann distribution:

p(s) ∝ e^−βH(s), (12)

where represents an admissible microstate of the cell aggregate and p(s) is the probability of the aggregate in the microstate s.

Based on these hypotheses, we consider a canonical ensemble of the system at a constant temperature T and upscale from the microscopic model described by the Hamiltonian (11) to a macroscopic coarse-grained model. Denote the collection of all microstates with prescribed average (Ē, [small xi, Greek, macron]) by

For a coarse-grained macroscopic model, we aim to calculate the effective free energy (per cell) with a prescribed average:where, by (12), the partition function with prescribed average is identified asWe remark that the effective free energy defined in (14) is such that the partition function without prescribed average is given bywhere the last equality justifies the terminology of “effective” free energy that governs the macroscopic coarse-grained properties of cell aggregates.

3. Effective free energy by mean-field approximation

For an N-cell aggregate in equilibrium, presumably cells would stay at the resting potential ξ* and tessellate the plane by regular hexagons to minimize the total energy. However, the presence of passive and active noises, size dispersion, topological defects, and thermal agitations give rise to variability and randomness in cell aggregates. From the postulated Boltzmann distribution (12), we can (i) identify the temperature or energy scale 1/β = k̃_BT from the fluctuations of states of cell aggregates, and (ii) predict the macroscopic measurable effective material properties (e.g., shear modulus and electromechanical coupling coefficients) and how they depend on the fluctuations or, equivalently, the energy scale k̃_BT. For these purposes, we need to evaluate the partition function (15) and the effective free energy (14) for some prescribed average/macroscopic strain and potential (Ē, [small xi, Greek, macron]).

3.1. Mean-field approximation

As for a standard Ising model, an exact solution to (15) is still an open problem for nontrivial interactions (encoded by the index set ). A widely accepted approximation method is the so-called the mean-field theory, which simplifies the problem by neglecting the contribution of interaction energy from fluctuations. More precisely, we rewrite the interaction energy of cell aggregates in (11) aswhere q is the number of neighboring cells (i.e. the size of the index set ) which is assumed to be independent of i. Denote by
the deviation from the average strain and average electric potential, respectively. In general, a Hamiltonian can be decomposed into three parts: the first depends only on the averaged or coarse-grained state variables, the second only on the deviations or high-frequency microscopic fluctuations, and the third on both. For instance, within the mean-field approximation (17) the Hamiltonian (11) of the cell aggregate can be written as

H(s) = H₀(s̄) + H₁(s′) + H₂(s′; s̄), (18)

where, by direct calculation, we find thatWe remark that the Hamiltonian (18), albeit anharmonic, is non-interacting because of the mean-field approximation. The associated partition function (15) and free energy (14) can be approximately evaluated by perturbation methods. To this end, we insert (18) and (19) into (14) and obtainFurther, upon neglecting the constraint in the integration domain, i.e., replacing by , we can write F̃ defined above as

Two remarks are worth noting here. First, the replacement of by can be justified a posteriori. As will be shown shortly, the expected value of the deviations in our subsequent calculations does satisfy 〈E′〉 = 〈ξ′〉 = 0. Second, the reduction of the integral over the high-dimensional space to is possible due to the mean-field approximation (18), which accounts for interactions between neighboring cells by mean-fields.

3.2. Variational perturbation method

We now evaluate the effective free energy (F^eff), i.e., the integral (21), using the variational perturbation method. However, the presence of anharmonic terms like |E′|⁴ and |E′|⁶ in Ψ (cf. (9)) defies an exact closed-form solution to (21). One approach to making progress is through the standard perturbation method. To this end, we choose a comparison energy function Ψ_c = Ψ_c(E′, ξ′) and denote bythe probability distribution function (PDF) associated with the comparison energy function Ψ_c. The statistical average using PDF (22) is defined as

Then the Bogoliubov inequality⁶³ asserts that for any energy function of E′ and ξ′, e.g., the function [small psi, Greek, tilde](E′, ξ′; Ē, [small xi, Greek, macron]) defined in (21), we have

F̃(Ē, [small xi, Greek, macron]; β) ≲ F̃^U(Ē, [small xi, Greek, macron]; β) = F_c + 〈[small psi, Greek, tilde](E′, ξ′; Ē, [small xi, Greek, macron]) − ψ_c(E′, ξ′)〉_c, (24)

where is a constant independent of coarse-grained state variables (Ē, [small xi, Greek, macron]).

We now analytically evaluate the right-hand-side of (24) by choosing a quadratic comparison energy function:

where μ_c, η_c > 0 and κ_c ∈ [Doublestruck R] are some constants that guarantee positivity of (25) and will be fixed later. For the best upper-bound estimate of F^eff, by (20) we may set

F^eff(Ē, [small xi, Greek, macron]; β) = Ψ(Ē, [small xi, Greek, macron]) + min{F̃^U(Ē, [small xi, Greek, macron]; β) : μ_c,η_c > 0, κ_c ∈ [Doublestruck R]}, (26)

where the minimization problem is over all reasonable Gaussian distribution (22) or positive harmonic comparison energy function ψ_c(E′, ξ′).

In this work, we consider a planar model in which the fluctuating strain E′, average strain Ē can be written as

The quadratic nature of the comparison energy turns integrals in (23) into Gaussian-type integrals which can be analytically evaluated. In particular, we are interested in evaluating the (Ē, [small xi, Greek, macron])-dependent terms on the right-hand side of (24), i.e., 〈[small psi, Greek, tilde](E′, ξ′; Ē, [small xi, Greek, macron])〉_c. From (9) and (21)₂, we find thatwhere OT(E′, ξ′; Ē, [small xi, Greek, macron]) represents terms that are odd in (E′, ξ′), and hence will not contribute to 〈[small psi, Greek, tilde]〉_c. By tedious but straightforward calculations, we find thatwhere μ₀ includes all terms that are independent of (Ē, [small xi, Greek, macron]), andand

Further, we find that the fourth-ordetensor satisfies that

meaning that the fourth-order tensor is isotropic and such thatInserting (30)–(32) into (29), we obtain the closed-form upper bound estimate for the effective free energy F^eff(Ē, [small xi, Greek, macron]; β) defined in (26) aswhere higher-order terms beyond and immaterial constants are neglected, and the relevant effective properties are given byWe remark that the effective free energy function, as defined in (33), governs the macroscopic electromechanical properties of cell aggregates.

4. Results and discussion

In this section, we explore biophysical implications of the coarse-grained model (33), with a particular focus on understanding how fluctuations impact the electromechanical behavior of cell aggregates. To facilitate meaningful comparisons with experimental results and numerical simulations, we select relevant model parameters in (9) such that their values lie within the physiological range. Specifically, the resting potential is chosen as ξ* = −60 mV,^12,64 the range of change in electric potential of a single cell as ([small xi, Greek, macron] − ξ*) ∈ [−200 mV, 200 mV],^13,16,17,64 the thickness of cell membrane as λ = 5 nm,⁶⁵ shear modulus as μ = 200 Pa,^66–69 the relative permittivity of a single cell as ε_r = 20,⁶⁵ and the electromechanical coupling constant for a single cell is chosen as . For simplicity, we focus on scenarios of simple shear with
Upon neglecting an (Ē, [small xi, Greek, macron])-independent constant, we rewrite the effective free energy function (33) as

4.1. Characterization of relevant energy scale

As demonstrated in ref. 55 and 57, the fluctuation 〈|E′|²〉_c can be directly obtained from experimental and numerical images of tissues. By (31), we see that the relevant energy scaleAside from a proportionality constant, the fluctuation 〈|E′|²〉_c plays exactly the same role as the temperature in the classical statistical mechanics and is referred to as the E²-temperature in ref. 55. E²-temperature characterizes stochasticity (or fluctuations) in cellular strains contributed by complex active and passive biological processes. Therefore, E²-temperature should be understood as some kind of empirical parameter or effective temperature^61,70–72 instead of the actual thermal temperature that originates from the vibration of atoms in classical thermodynamics. Subsequently, we study how macroscopic electromechanical properties of the cell aggregate depend on the E²-temperature (or fluctuation).

4.2. Renormalized resting potential

At the absence of external mechanical loading, the equilibrium state of the cell aggregate is determined by the principle of minimum free energy:It is easy to see that a local small-strain minimizer is given byDepending on the sign of K in the single-cell Hamiltonian (9), the E²-temperature 〈|E′|²〉_c effectively lowers (if K > 0) or increases (if K < 0) the resting potential. We remark that this effective resting potential ξ*^eff does not account for active processes of living cells. Biological processes, including ionic transport through channels and pumps, metabolic activities, and other cellular interactions, can significantly alter the electrical state of the cell.

4.3. Solid-to-fluid phase transition

It has been widely shown that cell aggregates may undergo a phase transition from a rigid solid-like phase to a floppy fluid-like phase, depending on the degree of “deviations” from the regular hexagonal tessellation.^16,17,55,57 In the current setting, we interpret this transition as the critical point such that the small strain local minimizer (38) loses its stability, meaning that†By (34) and (38), we find the critical E²-temperature to be the one at which the cell aggregate at resting condition (i.e., no external shear stress), transits from a solid-like phase to a fluid-like phase asBi et al. (2016⁷³) show that the solid–fluid phase transition occurs at the critical shape index p̂_cr = 3.81 at the absence of electromechanical coupling. By (5) we may fix γ* ≈ 2.0 for consistency.

Moreover, living cells are capable of regulating their transmembrane potentials. At the presence of electromechanical coupling, tissues can actively regulate their stiffness and solid-to-fluid transition by manipulating their electric potentials. This capability allows them to adaptively respond to varying environmental conditions and physiological demands. Suppose that the average electric potential of a cell aggregate is fixed at [small xi, Greek, macron] via certain active biophysical processes. By (35) we find that the apparent shear modulus as

The critical E²-temperature (or fluctuation) for solid–fluid phase transition at this prescribed potential [small xi, Greek, macron] is identified as that at which the apparent shear modulus μ^apparent of the cell aggregate vanishes:

Fig. 3 shows the phase diagram for solid-to-fluid phase transition governed by the coarse-grained model (35). The black solid line shows the phase boundary. The region below and above the phase boundary corresponds to the rigid solid-like phase and floppy fluid-like phase, respectively. At a fixed applied electric potential [small xi, Greek, macron], an increase in E²-temperature 〈|E′|²〉_c would transform the cell aggregate from a rigid solid-like phase to a floppy fluid-like phase. Similarly, at a prescribed E²-temperature, an increase in electric potential [small xi, Greek, macron] would transform the cell aggregate from a fluid-like phase to a solid-like phase.

Fig. 3 Phase diagram for solid-to-fluid phase transition in cell aggregates. For chosen numerical values in this work, the range of non-dimensional x-axis corresponds to the range of change in electric potential as ([small xi, Greek, macron] − ξ*) ∈ [−200 mV, 200 mV] consistent with the physiological range; the y-axis represents the E²-temperature 〈|E′|²〉_c.

4.4. Energy barrier

The effective free energy function (35) is anharmonic and admits multiple critical points in γ for a prescribed average potential [small xi, Greek, macron]. Graphically, as γ increases from 0, we observe that the first critical point (γ = 0) and the third critical point (γ_T1) are locally stable while the second critical point is unstable, reflecting the T1-transition illustrated in Fig. 2. To find the energy barrier between the two stable equilibria, we consider the equations for critical points:
By (35) we find that the unstable small-strain solution to the above equation is approximately given by
Therefore, by (35) the energy barrier between the stable equilibria γ = 0 and γ = γ_T1 is given by
The energy barrier can be used to quantitatively assess the rate of T1-transition. Following the classical Kramer's model,⁷⁴ we may postulate the rate k of T1-transition per unit volume aswhere the positive dimensionless constant follows from the last equality in (31). We remark that (43) is consistent with the physical interpretation of solid-to-fluid phase transitions determined by (41). In particular, the transition rate k ≪ 1 if the fluctuation 〈|E′|²〉_c ≪ 1 whereas k ∼ 1 if μ^apparent = 0. Moreover, the rate of transition may be identified as the self-diffusivity which, by the Stokes–Einstein's relation, implies that the viscosity η of the cell aggregates should satisfy

4.5. Electromechanical responses

Finally, we consider the electromechanical properties of the cell aggregate in the solid phase. Suppose that the cell aggregate is under the application of external shear stress σ^e. By (35), we observe that the external shear stress can alter the average cross-membrane potential [small xi, Greek, macron]. Conversely, changes of the average transmembrane potential [small xi, Greek, macron] can influence the overall shear strain of the cell aggregate. These interactions highlight a two-way coupling between mechanical and electrical responses within the system. Furthermore, both responses are significantly modulated by E²-temperature (or fluctuations) of the system. For a fixed applied shear stress σ^e, average potential [small xi, Greek, macron], and E²-temperature, we obtain the equilibrium shear strain γ^eq by the principle of minimum free energy:That is, γ^eq is the solution of the following nonlinear algebraic equation (cf. (35)):In general, the above nonlinear equation admits multiple solutions. We are only interested in the “small-strain” solution, which we numerically compute using MATLAB.

We first consider the dependence of strain on the external stress under different E²-temperature 〈|E′|²〉_c. As an example, we set the average electric potential the same as the effective resting potential ξ*^eff in (38): [small xi, Greek, macron] = ξ*^eff. Fig. 4 shows the stress–strain relation of cell aggregate at different E²-temperatures (or fluctuations) below the critical E²-temperature in (40). We observe that the macroscopic elastic response of the cell aggregate is generally nonlinear and matches the typical range for cell aggregates.⁶⁸ We also notice that an increase in E²-temperature (or fluctuation) softens the macroscopic elastic response, consistent with the effect of thermal temperature for classical crystalline solids.

Fig. 4 Macroscopic non-linear elastic response of cell aggregate for known fluctuation and applied electric potential [small xi, Greek, macron] = ξ*^eff.

Next, we consider the dependence of strain on the average cellular potential. Fig. 5 shows the equilibrium shear strain γ^eq plotted against normalized applied electric potential for σ^e = 0.02μ and different E² -temperatures 〈|E′|²〉_c. We observe that as the average electric potential increases, the equilibrium shear strain decreases nonlinearly, showing a nonlinear piezoelectric response for cell aggregates. For a given average electric potential, the equilibrium shear strain is higher for higher E²-temperature. We also observe that the changes in equilibrium shear strain due to changes in average electric potential are more significant at higher E²-temperature. Physically, this implies that a softer cell aggregate possesses an enhanced electro-mechanical response, making its shape changes more significant and more sensitive to the changes in transmembrane electric potential. This observation is consistent with the solid-to-fluid phase transition for cell aggregate (cf. Fig. 3) that for a given average electric potential, an increase in E²-temperature (or fluctuation) brings a solid cell aggregate closer to the solid-to-fluid phase transition point.

Fig. 5 Macroscopic inverse piezoelectric-like response of cell aggregate. Equilibrium shear strain γ^eq is plotted against normalized applied electric potential along x-axis. For chosen numerical values in this work, the range of non-dimensional x-axis in the plot corresponds to the range of change in electric potential as ([small xi, Greek, macron] − ξ*) ∈ [−200 mV, 200 mV] consistent with the physiological range.

5. Concluding remarks

In this work we first established the concept of strain tensor that characterizes the shape and orientation of individual cells and local arrangement of cells in amorphous cell aggregates. Based on the underlying structural symmetry and T1-transition experimentally observed in tissues, we proposed an electromechanical model for single cells and their interactions. For collective behavior of cell aggregates, we employed the approach of statistical mechanics and achieved a coarse-grained model for macroscopic electromechanical properties of cell aggregates. Fluctuation in cellular strains can be precisely quantified, which is referred as the E²-temperature for its analogous role as the absolute temperature in a conventional statistical model. The closed-form expression of the effective free energy enabled us to predict the macroscopic electromechanical properties of the cell aggregate. In particular, we derived the renormalized mechanical and electromechanical coupling coefficients and highlight how the average electric potential and E²-temperature can independently regulate the electromechanical properties and solid–fluid phase transitions in cell aggregates. Our work resolves the controversy of whether soft biological tissues are piezoelectric. The answer is simple. They are not intrinsically piezoelectric since they lack an underlying polar structure however, due to the electrostriction (or Maxwell stress) behavior of a single cell, and the resting potential, the aggregate or collective behavior of cells can mimic piezoelectric-like behavior.

There are several possible future directions: (i) the developed theory for living cell aggregates could be extended to explicitly account for the active processes (e.g., effects of active protein forces) by using the principles of non-equilibrium statistical mechanics.^50,75,76 (ii) The derived effective free-energy may be used to explore the rich instability and bifurcation behavior in soft tissues cf. ref. 77 and 78 (iii) We have ignored the electromechanical coupling mechanism of flexoelectricity⁶ or rather its possible emergence at the tissue scale. Prior work appears to indicate that flexoelectricity, at least at the cellular level, is implicated in a variety of biophysical phenomena such as the hearing mechanism cf. ref. 79–81, and may also have consequences for energy extraction at the coarser level cf. ref. 82. An approach similar to outlined in this work may be used to address this mechanism. (iv) The developed theory can be augmented to include the non-local inter-cellular interactions^83–87 by going beyond the nearest-neighbor interaction assumption used in this work.

Data availability

All data used in the paper is included within the paper.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

L. L. gratefully acknowledges the support of NSF DMS-2306254 and Simons travel grant. P. S. gratefully acknowledges the support of the Hugh Roy and Lillie Cranz Cullen Distinguished University Chair from the University of Houston.

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Footnote

† We assume that the phase-transition is of the 2nd order type. Some evidence supports a first-order type transition. Our framework may be extended to address the latter scenario if desired.

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