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Roth, Tan, and van Wijnen: A mathematical model of mechanotransduction

Review article

A mathematical model of mechanotransduction

Bradley J. Roth [1]

Author Affiliations

¹Department of Physics, Oakland University, Rochester, MI, USA.

Academic Editor(s):

Cheemeng Tan

Andre van Wijnen

Cite paper

Roth BJ. A mathematical model of mechanotransduction. Academia Biology 2023;1. https://doi.org/10.20935/AcadBiol6081

DOI: https://doi.org/10.20935/AcadBiol6081

Received: 15 Dec 2022,

Accepted: 31 Jan 2023,

Published: 22 Mar 2023

Open Access

Abstract

This article reviews the mechanical bidomain model, a mathematical description of how the extracellular matrix and intracellular cytoskeleton of cardiac tissue are coupled by integrin membrane proteins. The fundamental hypothesis is that the difference between the intracellular and extracellular displacements drives mechanotransduction. A one-dimensional example illustrates the model, which is then extended to two or three dimensions. In a few cases, the bidomain equations can be solved analytically, demonstrating how tissue motion can be divided into two parts: monodomain displacements that are the same in both spaces and therefore do not contribute to mechanotransduction, and bidomain displacements that cause mechanotransduction. The model contains a length constant that depends on the intracellular and extracellular shear moduli and the integrin spring constant. Bidomain effects often occur within a few length constants of the tissue edge. Unequal anisotropy ratios in the intra- and extracellular spaces can modulate mechanotransduction. Insight into model predictions is supplied by simple analytical examples, such as the shearing of a slab of cardiac tissue or the contraction of a tissue sheet. Computational methods for solving the model equations are described, and precursors to the model are reviewed. Potential applications are discussed, such as predicting growth and remodeling in the diseased heart, analyzing stretch-induced arrhythmias, modeling shear forces in a vessel caused by blood flow, examining the role of mechanical forces in engineered sheets of tissue, studying differentiation in colonies of stem cells, and characterizing the response to localized forces applied to nanoparticles.

1. Introduction

Suppose your goal is to create a mathematical model of mechanotransduction. What would that model look like? It might resemble the mechanical bidomain model [1]. You would want to include a crucial feature that many depictions of tissue biomechanics lack: the interaction of the intracellular cytoskeleton and extracellular matrix through integrin proteins in the cell membrane. The bidomain model predicts where mechanotransduction happens. This macroscopic model describes tissue averaged over many cells, ignoring its microscopic cellular structure. It can analyze diverse phenomena, such as heart remodeling, the production of engineered cardiac tissue, and stem cell differentiation. The objectives of this review are to introduce the mechanical bidomain model, survey its uses, and explore its applications.

In 1999, Matthias Chiquet [2] wrote,

Integrins … physically link the ECM [extracellular matrix] to the cytoskeleton, and hence are responsible for establishing a mechanical continuum by which forces are transmitted between the outside and the inside of cells in both directions … Because of their strategic location, integrins are good candidates for sensing changes in tensile stress at the cell surface … There is evidence that upon mechanical stimulation via the ECM, integrins (or associated proteins) could trigger signals which lead to adaptive cellular responses.

Chiquet claimed that integrins initiate a cascade of molecular events that lead to mechanotransduction. A similar role for integrins has been proposed by several researchers [3–11]. Let us illustrate this concept with pictures before we translate it into mathematics. Figure 1 portrays how integrins (red) bridge the extracellular matrix (blue) and the cytoskeleton (green). What triggers the integrin’s response? If the displacement of the extracellular space, w, differs from that of the intracellular space, u, then the two ends of the integrin would be pulled apart, causing it to deform. The fundamental hypothesis of the mechanical bidomain model is that the difference of the two displacements, u – w, causes mechanotransduction.

Figure 1

The extracellular matrix (blue) interacting with the cytoskeleton (green) through an integrin protein in the cell membrane (red). Differences in the intra- and extracellular displacements, u and w, distort the integrin. The underlying hypothesis of the mechanical bidomain model is that such distortions cause mechanotransduction.

2. Derivation of equations governing the one-dimensional bidomain model

2.1. The equations for a one-dimensional bidomain

Figure 1 emphasizes two elements required for any mathematical model of mechanotransduction: (1) separate motion in the intracellular and extracellular spaces, and (2) a way to join the two spaces by integrins. Consider a one-dimensional example. Assume the extracellular matrix is elastic so we can represent it by a line of springs (blue in Figure 2). We also represent the cytoskeleton as a line of springs (green). This picture of the intracellular space is not obvious because tissue consists of individual cells. For our model to make sense, cells must be connected by junctions called adhesions, so if you pull one cell, the force is transferred to adjoining cells. Finally, we represent the integrins as springs connecting the two spaces (red). Figure 2 shows the result: a ladder of springs.

To cast this model mathematically, assume the extracellular stress τ_e is proportional to the extracellular strain ε_e, so that τ_e = με_e, where μ is the extracellular mechanical modulus and the strain is the spatial derivative of the displacement, ε_e = dw / dx. Likewise, the intracellular stress is proportional to the intracellular strain, τ_i = νdu / dx, where ν is the intracellular modulus. The tissue is in equilibrium: the sum of the forces is zero. The force at any location arises from the difference between the stresses on the left and right (in other words, the derivative of the stress) and from the pull of the integrins. We interpret the integrins to be Hookean springs with spring constant K. The integrin term is responsible for linking the two spaces and is proportional to the difference between the displacements inside and outside the cells, u−w:

\frac{d τ_{i}}{d x} = K (u - w),

(1)

\frac{d τ_{e}}{d x} = - K (u - w) .

(2)

Figure 2

The one-dimensional mechanical bidomain model. Green springs represent the intracellular space, blue springs the extracellular space, and red springs the integrins in the membrane.

If we combine the stress–strain relationships with the definitions of the strain as derivatives of the displacements, and insert them all into equations 1 and 2, the expressions for mechanical equilibrium become

ν \frac{d^{2} u}{d x^{2}} = K (u - w),

(3)

μ \frac{d^{2} w}{d x^{2}} = - K (u - w) .

(4)

Equations 3 and 4 define the one-dimensional mechanical bidomain model. The term “bidomain” implies we consider two (“bi-”) spaces (“-domains”): intracellular and extracellular. The term “mechanical” distinguishes this model from the more familiar electrical bidomain model [12]. We stress “one-dimensional” because, as we will soon see, the model can be generalized to two or three dimensions.

Equations 3 and 4 are a pair of coupled differential equations. To better comprehend their nature, consider what happens when you add them. The right-hand sides have opposite signs (Newton’s third law), so they cancel and you get

\frac{d^{2}}{d x^{2}} (ν u + μ w) = 0 .

(5)

Equation 5 governs the behavior of a “monodomain.” It represents a single line of springs with a displacement expressed as a weighted combination of the displacements inside and outside the cells. The integrin spring constant K does not appear in Equation 5, so it does not influence the monodomain behavior.

Next, divide Equation 3 by ν and Equation 4 by μ and then subtract. The result is

\frac{d^{2}}{d x^{2}} (u - w) = (\frac{1}{υ} + \frac{1}{μ}) K (u - w) .

(6)

We call this the “bidomain” equation for $u - w$ . Our fundamental hypothesis is that the difference in displacements drives mechanotransduction, so this equation predicts where mechanotransduction occurs. Its solution is an exponential having a length constant σ given by

σ = \sqrt{\frac{ν μ}{K (ν + μ)}},

(7)

which has units of length and regulates how rapidly the exponential falls off with distance. It is the primary parameter in the mechanical bidomain model. As the spring constant K increases (the spaces are more tightly linked), the length constant σ decreases (the bidomain displacement is confined more locally).

2.2. Comparing the mechanical bidomain model to the cable model of a nerve axon

Equation 6 is well known to those who have studied bioelectricity; it is the one-dimensional cable equation describing current and voltage along a nerve axon,

\frac{d^{2} V_{m}}{d x^{2}} = \frac{V_{m}}{λ^{2}},

(8)

where V_m is the transmembrane potential—the difference between the intra- and extracellular voltages—and λ is the electrical length constant, which depends on the resistances inside and outside the axon and the conductance of the membrane. The electrical and mechanical bidomain models are similar [1]: electrical potentials are analogous to mechanical displacements, electrical conductivities are analogous to mechanical moduli, electrical current densities are analogous to mechanical stresses, and the electrical length constant λ is analogous to the mechanical length constant σ. In the electrical model, ion channels in the membrane open and close depending on the transmembrane potential. In the mechanical model, integrin proteins in the membrane activate and inactivate depending on the difference between displacements across the membrane, the “transmembrane displacement.”

3. Extension of the bidomain model to two dimensions

3.1. Extending the mechanical bidomain model to two dimensions

We can extend the mechanical bidomain model to two or three dimensions. For instance, in the two-dimensional model shown in Figure 3, the extracellular matrix (blue) and cytoskeleton (green) are represented by grids of springs coupled by integrins (red).

Figure 3

The two-dimensional mechanical bidomain model. Green springs represent the cytoskeleton, blue springs the extracellular matrix, and red springs the integrins.

In this case, the stress–strain relationships are more complicated than in the one-dimensional model because the elastic properties of a material are characterized by two parameters: the shear and bulk moduli [13]. Tissue is mostly water, which is nearly incompressible. Therefore, we can define the hydrostatic pressure [14, 15] as the product of a minuscule volume change and a gigantic bulk modulus. The intracellular stress is represented by the components of a two-dimensional matrix, or tensor

τ_{ixx} = - p + 2 ν ε_{ixx}, τ_{iyy} = - p + 2 ν ε_{iyy}, τ_{ixy} = 2 ν ε_{ixy},

(9)

where p is the pressure and ν is the shear modulus. The extracellular space is similarly

τ_{exx} = - q + 2 μ ε_{exx}, τ_{eyy} = - q + 2 μ ε_{eyy}, τ_{exy} = 2 μ ε_{exy},

(10)

where q is its pressure and μ is its shear modulus. With these stress–strain relationships, the standard strain–displacement relationships [16, 17], and the assumption of incompressibility, the mechanical bidomain equations become

- \frac{\partial p}{\partial x} + ν (\frac{\partial^{2} u_{x}}{\partial x^{2}} + \frac{\partial^{2} u_{x}}{\partial y^{2}}) = K (u_{x} - w_{x}),

(11)

- \frac{\partial p}{\partial y} + ν (\frac{\partial^{2} u_{y}}{\partial x^{2}} + \frac{\partial^{2} u_{y}}{\partial y^{2}}) = K (u_{y} - w_{y}),

(12)

- \frac{\partial q}{\partial x} + μ (\frac{\partial^{2} w_{x}}{\partial x^{2}} + \frac{\partial^{2} w_{x}}{\partial y^{2}}) = - K (u_{x} - w_{x}),

(13)

- \frac{\partial q}{\partial y} + μ (\frac{\partial^{2} w_{y}}{\partial x^{2}} + \frac{\partial^{2} w_{y}}{\partial y^{2}}) = - K (u_{y} - w_{y}) .

(14)

The first two equations govern the cytoskeleton, and the second two govern the extracellular matrix. The first and third equations govern forces in the x direction, and the second and fourth equations govern forces in the y direction.

3.2. Incompressibility

The incompressibility of the tissue means that u and w have zero divergence

\frac{\partial u_{x}}{\partial x} + \frac{\partial u_{y}}{\partial y} = 0, \frac{\partial w_{x}}{\partial x} + \frac{\partial w_{y}}{\partial y} = 0.

(15)

In previous research based on the two-dimensional bidomain model, we employed stream functions to ensure incompressibility [16, 17]. We will not do that here, but in some situations, they simplify the analysis. In a three-dimensional calculation, stream functions are replaced by vector potentials [18].

4. Analytical predictions of the model

Originally, the mechanical bidomain model was derived by Puwal and Roth [19] to describe magnetic forces. A bidomain model was necessary because the magnetic force is the product of the current and the magnetic field, and the currents inside and outside the cells that are associated with a propagating action potential are usually equal but opposite. Therefore, the intra- and extracellular forces cancel, so the net magnetic force acting on the tissue vanishes. Nevertheless, the cytoskeleton is pushed in one direction and the extracellular matrix is pulled in the other, producing opposing motions that are a distinguishing feature of bidomain behavior. It turned out that cardiologists do not care about magnetic forces, but our analysis of them led to the mechanical bidomain model. Later, it became evident that the bidomain model could describe mechanotransduction.

4.1. Analytical analysis of shearing a slab of cardiac tissue

One of the first predictions of the bidomain model arose from analyzing a slab of tissue that is sheared [20]: the upper surface is pulled right, and the bottom surface is pulled left (Figure 4). The displacement is split into two parts. The monodomain part (a weighted sum of the inside and outside displacements, representing the average motion of the entire tissue) varies linearly across the slab, producing a uniform shear strain. The bidomain part (the difference between the inside and outside displacements) decays exponentially from the upper and lower surfaces. If the length constant σ is much less than the slab thickness, then the bidomain term is negligible everywhere except near the surfaces. In general, the magnitude of the bidomain displacement is smaller than the monodomain one, so in Figure 4 (and in other figures), we exaggerate the bidomain arrows so that they are discernible. If mechanotransduction occurs in this slab of tissue, it will be localized to within a few length constants of the upper and lower surfaces.

4.2. Analytical analysis of contraction in a sheet of tissue

When muscle contracts, we must add an additional term to the intracellular stress to represent the active tension T developed by actin and myosin [14, 15],

τ_{ixx} = - p + 2 ν ε_{ixx} + T,

(16)

where we assume that the muscle fibers are straight and run along the x axis. Roth [21] analyzed a circular sheet of active cardiac tissue (Figure 5). The sheet contracts along the fibers and incompressibility causes it to expand across to the fibers. In this case, the mathematical analysis is complex because the equations must be expressed in polar coordinates and the solution contains modified Bessel functions. Nevertheless, like before the displacement is divided into two parts: monodomain and bidomain. The monodomain strain is dispersed throughout the tissue. The bidomain displacement, however, is restricted to a layer near the tissue edge whose width is set by the length constant σ. Mechanotransduction occurs only in this boundary layer.

The pressures do not vanish for the example shown in Figure 5 (they did for case examined in Figure 4), and the bottom panels in Figure 5 depict the distributions of intra- and extracellular pressure. The pressure is difficult to interpret. First, the bidomain pressure is a macroscopic quantity that may not be equivalent to the microscopic pressure [22]. Second, a difference between the intra- and extracellular pressures could cause water to flow into or out of the cells, changing their volume. The calculation in Figure 5 assumes that the contraction occurs so quickly that water does not have time to redistribute between the two spaces.

Figure 4

The monodomain and bidomain displacements in a slab of sheared cardiac tissue; the top is pulled right and the bottom left. The sizes of the bidomain arrows are exaggerated.

Figure 5

Top: Contraction of a circular sheet of cardiac tissue. Red lines show the fiber direction, which is horizontal in the monodomain and bidomain panels. The dotted oval in the monodomain panel reveals how the sheet deforms when fibers contract. Bottom: The intra- and extracellular pressures, p and q. The pressures are normalized by their maximum value.

The reason pressures arise is because both spaces are incompressible. To explore incompressibility further, Sharma and Roth [22] altered the model to contain both shear and bulk moduli in each space. Because the bulk modulus allows for variation in volume, the tissue was compressible. They examined several cases, including a reanalysis of the sheet of cardiac tissue shown in Figure 5. Making the tissue compressible did not influence the displacements markedly but did affect the pressures. In addition, it introduced a second length constant, similar to the original one (Equation 7) except that it contained the bulk rather than the shear moduli. The pressure distributions were uniform throughout the tissue except near the edge where they varied over a few of the new length constants. Sharma and Roth estimate that the length constant containing the bulk moduli should be about three hundred times larger than the one containing the shear moduli. If the bulk-modulus length constant is much larger than the dimensions of the tissue slab, the results of the compressible and the incompressible models are nearly identical.

5. Insight into the behavior of the mechanical bidomain model

5.1. A tug-of-war analogy for the bidomain model

Both Figures 4 and 5 contain a boundary layer of bidomain displacement. This layer arises mathematically from the form of Equation 6, but why does it emerge physically? When a force is applied to the tissue, it generates a stress equal to the force divided by the cross-sectional area. In the bidomain model, this stress is distributed between the intra- and extracellular spaces. For example, if the extracellular matrix is flexible but the intracellular cytoskeleton is stiff $(μ ≫ ν)$ , then the stress outside the cells is much less than the stress inside the cells when the two spaces move in tandem. Near the tissue edge, however, the allocation of stresses is set by the boundary conditions. For instance, in Figure 4, the force F is applied to the extracellular matrix, while the intracellular cytoskeleton is stress-free. As you move into the interior of the tissue, the stress redistributes between the two spaces according to the relative magnitudes of their shear moduli ν and μ. Deep in the tissue, where redistribution is complete, the displacements and strains are identical in the two spaces, although the stresses differ. This redistribution of stresses takes place over a few length constants.

This analysis of stresses is similar to the “tug-of-war” concept introduced by Trepat and Fredberg [23], which describes the forces shown in Figure 6. The forces on the rope are like those in the intracellular space (green), the forces on the ground are like those in the extracellular space (blue), and the forces between the people’s feet and the ground are like those acting on integrins (red). The illustration differs from that of Trepat and Fredberg because of the addition of forces on the ground; it is as if tug-of-war were being played on a flexible surface like a trampoline. This figure highlights three critical attributes of the model: (1) the significance of the relative size of the intra- and extracellular shear moduli, (2) the role of the length constant in redistributing the stresses, and (3) the impact of boundary conditions on the model predictions.

Figure 6

Top: The “tug-of-war” model of tissue biomechanics. The cytoskeleton (green), extracellular matrix (blue) and integrin (red) forces acting in tissue, represented as forces interacting between people playing tug-of-war. Bottom: Representation of the mechanical bidomain model by a ladder of springs, as in Figure 2 except flipped so the cytoskeleton is on top. The tug-of-war picture is modified from a photo on Wikipedia (https://upload.wikimedia.org/wikipedia/commons/a/a3/Irish_600kg_euro_chap_2009_%28cropped%29.JPG, GNU Free Document License, https://commons.wikimedia.org/wiki/Commons:GNU_Free_Documentation_License,_version_1.2).

5.2. Comparison of the electrical and mechanical bidomain models

One hallmark of the electrical bidomain model is the “unequal anisotropy ratios.” In cardiac tissue, the intra- and extracellular electrical conductivities are anisotropic: they are about the same size parallel to the myocardial fibers, but the intracellular conductivity is less than the extracellular conductivity perpendicular to the fibers [1]. An analogous feature arises in the mechanical bidomain model [17]. Mechanical moduli also can be anisotropic, and the anisotropy may not be the same in the two spaces. Earlier we mentioned how important are the relative values of the intra- and extracellular shear moduli during the redistribution of stresses in tissue. If the mechanical moduli have unequal anisotropy ratios, this may lead to intriguing and unexpected behaviors as stresses redistribute between the intra- and extracellular spaces where fibers turn. Curving and rotating fiber geometries, which are often encountered in the heart [24, 25], may cause mechanotransduction “hot spots” [16, 17]. Cardiac monolayers can be created with any user-specified fiber geometry [26–28] and could provide a sensitive way to test this prediction.

A common experiment in biomechanics is indentation [29–32], where a miniature probe pushes down on the tissue surface. If the probe is in direct contact with the extracellular matrix, the stresses redistribute into the cytoplasm over a few length constants from the probe tip, and we anticipate a region where there are significant differences between u and w, producing mechanotransduction. However, unequal anisotropy ratios may cause this redistribution to have a surprising spatial pattern [18], like the unanticipated spatial distribution of transmembrane potential around a stimulating electrode predicted by the electrical bidomain model [1].

Another unexplored aspect of the mechanical bidomain model is the relationship between the macroscopic model and the microscopic tissue structure. Again a parallel exists between mechanical and electrical models. One hypothesis for how electric shocks affect the heart is the “sawtooth model,” a microscopic model that separates the electrical resistance of the cytoplasm from the resistance of the gap junctions connecting cells [33, 34]. Gap junctions in the electrical model are analogous to adhesions (mechanical intercellular junctions) in the mechanical model. Forces acting on adhesions may lead to mechanotransduction [35]. Such considerations extend beyond the macroscopic bidomain model and explore the macroscopic/microscopic relationship. Mertz et al. [36] found that biomechanical behavior is sensitive to cell–cell adhesions, and this sensitivity might provide another tool to probe mechanotransduction.

One difference between the electrical and mechanical models is that the transmembrane potential is a scalar (it has a magnitude but no direction), whereas the bidomain displacement is a vector (it has both a magnitude and a direction). Moreover, in the electrical model, a positive transmembrane potential (depolarization) has a different outcome than a negative transmembrane potential (hyperpolarization). Our working hypothesis is that the magnitude |u – w| is the key quantity in the mechanical bidomain model, not its sign or direction. But we do not know this for sure, and maybe its direction is important too.

6. Numerical calculations using the model

6.1. Using perturbation theory to analyze the bidomain model

Just a few biomechanics problems can be solved analytically; most require numerical analysis. Punal and Roth [37] analyzed the mechanical bidomain model using perturbation theory [38]. If a distance characterizes the tissue, such as the thickness of the slab (Figure 4) or the radius of the sheet (Figure 5), we can divide the bidomain length constant σ by this characteristic distance to form a dimensionless parameter. In most cases, we expect that σ will be much smaller than these other lengths, so this dimensionless parameter will be tiny. Punal and Roth expanded their mathematical expressions in powers of this parameter and then collected terms with like powers. Their zeroth-order expressions governed the lowest-order monodomain contribution, and the first-order expressions governed the first nonzero bidomain contribution. Only the bidomain term causes mechanotransduction (the monodomain term corresponds to identical movement in the two spaces so it does not contribute to their difference). Thus, perturbation theory suggests a two-step process: first solve the monodomain equation just like everyone else in the field of biomechanics does, and then use this solution and the first-order equation to calculate the bidomain contribution. This technique would be beneficial because it would demonstrate how the bidomain behavior could be obtained from monodomain calculations that preceded it [39–42]. Monodomain models often contain important elements not yet incorporated into the bidomain model, such as large strains, nonlinear stress–strain relationships, complicated fiber geometries, and irregular tissue boundaries. Perturbation methods may furnish a way to compute bidomain displacements from previous sophisticated and nonlinear monodomain simulations.

6.2. Solving the bidomain equations using computational methods

Another approach to analyzing the mechanical bidomain model is to solve the full equations numerically on a computer using a computational technique like the finite difference method. Gandhi and Roth [43] developed such a method to investigate remodeling of tissue around an ischemic region in the heart (Figure 7). The central circular area is deprived of oxygen and cannot contract. The surrounding tissue is healthy and can generate a uniform tension T acting along the myocardial fibers (horizontal). When the healthy tissue contracts, it stretches the ischemic region; because the tissue is incompressible, this region is flattened perpendicular to the fibers. A wide-spread distribution of monodomain strain is present throughout both the ischemic area and the surrounding healthy tissue. The bidomain displacement, however, is confined to the ischemic region’s border zone. The calculation predicts that remodeling of cardiac tissue—a type of mechanotransduction—should occur primarily in the border zone, consistent with observations [44].

Sharma et al. [20] performed the first bidomain calculation using the more powerful finite element method. Such calculations are important because the finite element technique is needed for tissues with realistic shapes and complicated fiber geometries [39, 45].

Figure 7

A sheet of cardiac tissue with a central ischemic region that cannot develop an active tension. When the surrounding healthy tissue contracts, the resulting displacement can be separated into monodomain and bidomain parts. The bidomain part, corresponding to the location of mechanotransduction, is limited to the border zone between the healthy and ischemic regions.

Figure 8

The multiscale nature of the mechanical bidomain model. The length scales range from the molecular (an integrin in the cell membrane) to the cellular (cardiac cells embedded in extracellular matrix) to the tissue (cardiac muscle with red lines showing the fiber direction) to the organ (a rabbit heart)

6.3. Multiscale models

The mechanical bidomain model is a multiscale model [46]: the disparate length scales extend from molecules to cells to tissue and finally to the whole organ, spanning spatial distances that differ by many orders of magnitude (Figure 8).

7. Precursors of the bidomain model

The mechanical bidomain model resembles models presented in earlier investigations. First and foremost, it is analogous to the electrical bidomain model [12], which is now the state-of-the-art method for simulating cardiac pacing and defibrillation [47]. The mechanical and electrical bidomain models have many similarities including their mathematical structure [1]. The mechanical bidomain model may become as valuable for studies of mechanotransduction as the electrical bidomain model is for studies of defibrillation.

7.1. Comparison of the bidomain model to biphasic models

Some mechanical models are called “biphasic” [48]. Of these, the best known is Mow’s biphasic model of cartilage [30, 49, 50]. The solid and fluid phases in cartilage are analogous to the cytoskeleton and extracellular matrix in cardiac tissue, and the frictional coupling of cartilage’s two phases is governed by a mathematical expression similar in form to the elastic coupling by integrins in the bidomain model.

7.2. Models of cell colonies that are similar to the bidomain model

The mechanical bidomain model mirrors those derived by Edwards and Schwarz [51] and Banerjee and Marchetti [52] to describe growing cell colonies [36, 53]. Edwards and Schwarz’s spring constant k is analogous to our constant K, and their localization length l corresponds to our length constant σ. However, they considered that the tissue adhered to a microstructured surface made up of an array of flexible pillars, like used in traction force experiments [54]. Our model, on the other hand, has the linking of the intra- and extracellular spaces occurring through integrins. Therefore, the significance of our coupling term is unlike that in previous models: in our model it is the signal that drives mechanotransduction.

8. Applications of the mechanical bidomain model

8.1. Growth and remodeling of the heart

The mechanical bidomain model has many potential applications. We already mentioned remodeling of cardiac tissue. Not only does the bidomain model predict tissue modifications in the border zone of an ischemic region, but also it might explain the thickening of the heart wall during hypertrophy [55]. Puwal [56] used the model to understand how the heart responds to high blood pressure. Many researchers [57–63] assume that factors such as ventricular wall stress are the stimuli for mechanotransduction, but our model is based on the assumption that differences of the motion inside and outside the cells drive these events. If this assumption is correct, a bidomain formulation may be essential for understanding remodeling.

8.2. Stretch-induced arrhythmias

The mechanical bidomain model might be able to forecast where stretch-activated ion channels open. In the heart, mechanoelectrical feedback [64] can cause stretch-induced arrhythmias [65] and influence defibrillation efficacy [66, 67]. We do not know the mechanism underlying stretch-activated ion channels; these channels may respond to membrane forces, or they may be controlled by stretch sensors inside the cells [68]. The bidomain model might help us distinguish between these two hypotheses.

8.3. Blood flow in vessels

Shear forces play an important role in the physiology of a blood vessel [69, 70]. The vascular endothelium is regulated by shear stress caused by blood flow [71], in part through the discharge of nitric oxide [72]. A model of a vessel and the blood flowing within it will enable us to examine the impact of the bidomain model on this behavior. Such simulations would require us to determine the appropriate boundary conditions connecting the tissue to the flow of blood.

8.4. Engineered cardiac tissue

Engineered tissue is becoming increasingly important for therapy [73–78]. Tissue engineering often requires manipulating mechanical forces [79–81]. In vitro tissue engineering relies on fabricating replacement tissue [82] grown on an extracellular matrix [83]. The mechanical stresses applied to growing engineered sheets of tissue influence its structure and function [84, 85]. For example, Fink et al. [86] stretched a sheet of engineered heart tissue and observed a greater concentration of cells growing at its edge. The mechanical bidomain model predicts the localization of tissue growth at an edge, a behavior that is characteristic of a bidomain [87].

8.5. Stem cell differentiation and development

Mechanical forces can control tissue growth [9] and stem cell differentiation [88–90]. In a cluster of cells, the environment at the periphery is often unlike that in the interior [52, 91–93]. For example, in a colony of growing human stem cells, the cells differentiate mainly at the border [94]. The mechanical bidomain model predicts that mechanotransduction occurs at the colony edge [95]. This data may be the most compelling yet in support of the bidomain approach. Moreover, Rosowski et al.’s experiment with growing colonies [94] estimates the size of the length constant σ. They observe edge effects that extend over about 150 microns, which is larger than the size of the individual cells but smaller than the radius of the colonies. If the bidomain model correctly describes the behavior of stem cell colonies, it might provide insight into complex processes during human development.

8.6. Local forces applied using magnetic nanoparticles

When exposed to a magnetic field, magnetic nanoparticles exert a localized force on cardiac tissue [96, 97]. In response to such a point force, the bidomain part of the tissue displacement falls off with distance more quickly than the monodomain displacement or strain [98], implying that tissue growth and remodeling is restricted to near where the force is applied.

8.7. The role of mechanotransduction in tumor biology and cancer therapy

Finally, evidence exists that integrins may play a role in tumor biology and cancer therapy [6, 99]. As a result, tumors might be influenced by bidomain behavior.

9. Conclusion

The mechanical bidomain model is still new. Some applications have not yet been analyzed, and the model may not prove fruitful in every case. Nevertheless, many researchers claim that integrins coupling the cytoskeleton to the extracellular matrix play a vital role in mechanotransduction. The mechanical bidomain model represents a first step in an attempt to describe this behavior mathematically [100]. Indeed, it is almost the simplest model one could derive that includes the linking of intra- and extracellular spaces by integrins. Additional features need to be added to the model when applying it to specific situations, but even in its most elementary form the model provides valuable insight into mechanotransduction. If the model predictions prove inconsistent with experiments, the process of deriving this model will still be valuable because it will force researchers to analyze why the model is incorrect. Several experiments—such as those studying stem cell colonies, growing sheets of engineered cardiac tissue, and analyzing remodeling in an ischemic border zone—are consistent with bidomain predictions. Additional experiments are needed to determine if the mechanical bidomain model will make a worthwhile contribution to mechanobiology.

Acknowledgments

The development of the mechanical bidomain model would not have been possible without the contributions of many excellent students, including Debabrata Auddya, Samip Gandhi, Vanessa Punal, Steffan Puwal, Kharananda Sharma, and Dilmini Wijesinghe.

Funding

The author declares no financial support for the research, authorship, or publication of this article.

Author contribution

The author confirms sole responsibility for this work. The author approved this work and takes responsibility for its integrity.

Conflict of interest

The author(s) declare no conflict of interest.

Data availability statement

Data supporting these findings are available within the article, at https://doi.org/10.20935/AcadBiol6081, or upon request.

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A mathematical model of mechanotransduction

Abstract

1. Introduction

Figure 1

2. Derivation of equations governing the one-dimensional bidomain model

2.1. The equations for a one-dimensional bidomain

(1)

(2)

Figure 2

The one-dimensional mechanical bidomain model. Green springs represent the intracellular space, blue springs the extracellular space, and red springs the integrins in the membrane.

(3)

(4)

(5)

(6)

(7)

2.2. Comparing the mechanical bidomain model to the cable model of a nerve axon

(8)

3. Extension of the bidomain model to two dimensions

3.1. Extending the mechanical bidomain model to two dimensions

Figure 3

The two-dimensional mechanical bidomain model. Green springs represent the cytoskeleton, blue springs the extracellular matrix, and red springs the integrins.

(9)

(10)

(11)

(12)

(13)

(14)

3.2. Incompressibility

(15)

4. Analytical predictions of the model

4.1. Analytical analysis of shearing a slab of cardiac tissue

4.2. Analytical analysis of contraction in a sheet of tissue

(16)

Figure 4

The monodomain and bidomain displacements in a slab of sheared cardiac tissue; the top is pulled right and the bottom left. The sizes of the bidomain arrows are exaggerated.

Figure 5

5. Insight into the behavior of the mechanical bidomain model

5.1. A tug-of-war analogy for the bidomain model

Figure 6

5.2. Comparison of the electrical and mechanical bidomain models

6. Numerical calculations using the model

6.1. Using perturbation theory to analyze the bidomain model

6.2. Solving the bidomain equations using computational methods

Figure 7

Figure 8

6.3. Multiscale models

7. Precursors of the bidomain model

7.1. Comparison of the bidomain model to biphasic models

7.2. Models of cell colonies that are similar to the bidomain model

8. Applications of the mechanical bidomain model

8.1. Growth and remodeling of the heart

8.2. Stretch-induced arrhythmias

8.3. Blood flow in vessels

8.4. Engineered cardiac tissue

8.5. Stem cell differentiation and development

8.6. Local forces applied using magnetic nanoparticles

8.7. The role of mechanotransduction in tumor biology and cancer therapy

9. Conclusion

Acknowledgments

Funding

Author contribution

Conflict of interest

Data availability statement

Institutional review board statement

Informed consent statement

Sample availability

Publisher’s note

References

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