Emergence of form from function—Mechanical engineering approaches to probe the role of stem cell mechanoadaptation in sealing cell fate

ABSTRACT

Stem cell “mechanomics” refers to the effect of mechanical cues on stem cell and matrix biology, where cell shape and fate are intrinsic manifestations of form and function. Before specialization, the stem cell itself serves as a sensor and actuator; its structure emerges from its local mechanical milieu as the cell adapts over time. Coupling of novel spatiotemporal imaging and computational methods allows for linking of the energy of adaptation to the structure, biology and mechanical function of the cell. Cutting edge imaging methods enable probing of mechanisms by which stem cells' emergent anisotropic architecture and fate commitment occurs. A novel cell-scale model provides a mechanistic framework to describe stem cell growth and remodeling through mechanical feedback; making use of a generalized virtual power principle, the model accounts for the rate of doing work or the rate of using energy to effect the work. This coupled approach provides a basis to elucidate mechanisms underlying the stem cell's innate capacity to adapt to mechanical stimuli as well as the role of mechanoadaptation in lineage commitment. An understanding of stem cell mechanoadaptation is key to deciphering lineage commitment, during prenatal development, postnatal wound healing, and engineering of tissues.

KEYWORDS:

• cell mechanics
• fate
• lineage commitment
• mechanoadaptation
• mechanomics
• stem cell

INTRODUCTION

Background

Structure – function relationships are ubiquitous throughout nature, across length and time scales. Recent studies indicate that stem cells are exquisitely sensitive to mechanical signals, which direct cell fate and thereby function. Specifically, stem cells exhibit more than a thousand fold greater capacity to sense and adapt to their local mechanical environment than differentiated cells of musculoskeletal and vascular lineages.^3-8 Specifically, stem cells upregulate transcription and secretion of structural proteins in response to short duration (minutes to an hour), subtle mechanical cues (1000x smaller in magnitude compared to those experienced by terminally differentiated cells). Such subtle mechanical cues arise e.g. from force transduction associated with cell proliferation and motility in early developmental periods, before the first beat of the heart or twitch of skeletal muscle.³ Cytoskeletal protein remodeling shows similar plasticity in response to changes in the stem cells prevailing mechanical environment.^6,8 Remarkably, the “astonishing… conserv[ation]” of the cytoskeletal actin filament “across a diverse set of eukaryotic species” implicates not only its ubiquity but also the utility of the cytoskeleton for “expansion of …functional diversity,”⁹ allowing for specialization of cell phenotype across species and within organisms.

While stem cells adapt to their dynamic local milieu, they also influence their local mechanical milieu directly through their very presence (Fig. 1).^10-12 In addition, stem cells modulate their own environment by altering their own structure as well as modulating emergent tissue architectures through up and down regulation of cytoskeletal, adhesion, and ECM protein transcription.^4,8 Although many published studies have addressed structure-function relationships in either terminally differentiated cells^5,6,13,14 or at mid-late stages of embryonic development where vascular pressure gradients^15,16 and/or muscle forces can be either measured or estimated,^3,17,18 very few studies have probed the mechanome at earliest stages of fate initiation or in live cells.^3-8

FIGURE 1. The cell itself and the ECM it generates modulate the cell's mechanical milieu at multiple length scales. (A) Transmission electron microscope image of an osteocyte process traversing the plane of the image and orthogonal to the plane in the right upper image half, superimposed with computational fluid dynamics predictions of pericellular flow at cell surfaces. In terminally differentiated osteocytes, the cell processes and local ECM amplify the transduction of mechanical cues via pericellular fluid flow. Color plot represents flow field, where v is the flow velocity. Velocity (m/s) increases at sites where ECM ingresses into the pericellular space. Used with permission.¹⁰ (B) Similar effects are observed around live model embryonic mesenchymal stem cells (C3H10T1/2, green) where flow fields are tracked using fluorescent microspheres (red). Used with permission.¹¹

The interplay of chemical and physical cues on stem cell differentiation comprises a current topic of intense study. The efficacy of differentiation media alone in driving specific stem cell fates is well established. While many studies have documented diverse cell behaviors in response to combinations of biochemical and biophysical cues, outcome measures depend strongly on experimental conditions and/or the labs applying them. For example, over the past decade, observations of the intertwined roles of cell shape and lineage commitment as well as the associated role of the substrate compliance on cell shape and lineage commitment have spawned scores of further studies on the same topic, with some indicating a correlation between substrate compliance and lineage commitment^22-24 and others indicating quite the opposite.^25,26 In short, while the definition of libraries including different combinations of mechanical and/or chemical cues that could be used prescriptively to guide targeted lineage commitment, and with spatial and temporal fidelity, could be widely applicatied to regenerative medicine, such libraries have yet to be created. Hence, the awe-inspiring question remains unanswered, “What fundamental principles drive the mechanome?,” where the term “mechanome” refers to biophysical cues that effect stem cell shape changes and emergent lineage commitment.¹⁹

Furthermore, few studies have noted that the non-motile (adherent) cell itself creates the ECM that makes up its own local environment and in this way significantly influences its own capacity to adapt and thereby to modulate ensuing lineage commitment (Fig. 2A). In contrast, extracellular factors in the motile cell's milieu drive its mechanoadaptation (Fig. 2B). Cell mechanoadaptation is achieved via cytoskeletal remodeling^29-33 as well as changes in cell-cell and cell-matrix adhesion complexes.¹⁰ Loss in junctional proteins favors a motile phenotype, enabling the cell to leave its current milieu and to become adherent in a new niche, where it may then lay down a matrix and become immotile once again.^3,33-35 Osteocytes and red blood cells may represent two terminally differentiated phenotypes where the cell is so highly specialized that the probability of transdifferentiation is very low if not statistically impossible.³

FIGURE 2. Relationship between motile and adherent stem cell states, MET/EMTs, and cell specialization through differentiation. The different states of the cell are depicted by a “cellular traffic light,” where motile states are depicted in green, adherent states in red, and transitional states in yellow. (A) Stem cells in adherent states may go through an MET or EMT to become motile. Osteocyte schematic used with permission.²⁷ (B) Once motile, circulating stem cells may again go through an MET or MET to become adherent.

Taken in context of epithelial to mesenchymal (EMT) and mesenchymal to epithelial transitions (MET), which serve as archetypes for cell mediated tissue genesis during prenatal development and postnatal healing, the genesis of ECM by stem cells may play a decisive role in sealing stem cell fate. Indeed, the very foundation of tissue genesis depends on initial arrangements of cells in space and time,¹⁰ resulting in patterning of the organism's template in prenatal development and a tissue template in postnatal healing. During embryogenesis, mesenchymal to epithelial and epithelial to mesenchymal transitions serve as hallmarks for emergence of new tissue types with unique tissue architectures. From a broad perspective, METs and EMTs provide a basis for switching between tissue architectures characterized by sheets and globular structures, where epithelia define boundaries or borders of cellular or tissue niches and mesenchyme defines the “interior milieu” or stuff therein. These paradigms for prototyping tissue templates are reiterated throughout development, growth, remodeling, and healing processes, in health (functional) and disease (dysfunctional) (Fig. 2).³

As noted by Waddington in his landscape of 1957, there are an infinite number of paths which a given pluripotent cell may take, and myriad environmental factors modulate the landscape topography.³⁶ Our working hypothesis is that the likelihood for one path to be taken versus another is determined by the state of the cell at that given point in time and corresponding energy landscape of the process. In other words, at any given time, the state of the cell is stochastic and the probability of any one behavior emerging is determined by energy cost of the process and energy supply of the cell. An understanding of the probability of one given outcome occurring vs. another requires an understanding of the physical state of the cell's machinery in relation to its environment, an understanding of the energy or work required to induce change, either through differentiation, proliferation or motility, as well as an understanding of the cell's own energy supplies.^37-40

Structure-function relationships at the cell's length and time scales

Already more than a century ago, D'Arcy Thompson “demonstrat[ed] that organic shapes conform to the physical forces prevailing at their scale…[and that] physical forces exert a direct and immediate influence in shaping organisms as they grow….”⁴¹ Further, in observing the structure-function relationships of cells and tissues, Thompson noted that “…it is in obedience to the laws of physics that [the cell's] particles have been moved, molded and conformed.” In describing these relationships at the length and time scale of the tissues comprising the femoral neck, he noted,

We must remember that our bone is not only a living but a highly plastic structure; the little trabeculae are constantly being formed, deformed, demolished, and formed anew. Here, for once, it is safe to say that ‘heredity’ need not and cannot be invoked to account for the configuration and arrangement of the trabeculae: for we can see than at any time of life in the making, under the direct action and control of the forces to which the system is exposed…Herein then lies, so far as we can discern it, a great part at least of the physical causation of what at first sight strikes us as purely physical adaptation: as a phenomenon, in other words, whose physical cause is as obscure as its final cause or end is apparently manifest.⁴¹

Without our current knowledge of stem cells and their regenerative capacity, Thompson articulated a current working hypothesis of mechanoadaptation, that cell lineage commitment (fate) evolves from the spatiotemporal enforcement of shape, via biophysical signals (e.g., mechanical) signals, over time. While the mechanoadaptation hypothesis did not gain favor in a biological community embracing Darwinism and natural selection of genetic traits which predispose “fitness to survive” on their beholders¹, it has served as a foundation for the growing field of mechanobiology. Recently, the set of mechanical cues relating to gene transcription profiles typical for specific lineages, the stem cell's “mechanome” has begun to be mapped (Fig. 3).^4,12,20,34

FIGURE 3. Retrospective mapping of the stem cells' mechanome, based on experimental data using an embryonic model mesenchymal stem cell line (C3H10T1/2). Actual stress and strain data points from experiments are represented as 95% confidence intervals, and color of resulting areas depicts the range of mechanical cues which correlating to early genetic markers of lineage commitment for chondrogenesis (yellow), chondrogenesis and haematopoesis (purple) and chondrogenesis, haematopoesis and osteogenesis (pink). In that sense, it maps states statistically conducive to fate. The next important step is to test prospectively whether those regions can be used to map libraries of mechanical cues to guide fate. Used with permission.²³

MECHANOCHEMISTRY OF STEM CELL ADAPTATION

Stem cell mechanoadaptation is an emergent property that is driven by the balance of prevailing mechanical forces at boundaries of the cell (plasma membrane) and the nucleus (nuclear membrane) as well as the prevailing biochemical supply and demand for raw materials necessary to adapt (which is highly influenced by transcription of genes that code for these raw materials including cytoskeletal, ECM and adhesion proteins).¹⁰ The capacity to act, i.e. to use these materials (or even to modulate their production) for adaptive purposes, relates to the energy state of the cell and the relative cost of adaptation. In this sense, the mechanoadaptation hypothesis regards stem cell fate as a manifestation of persistent structure-function relationships, a reaching of steady state enabled by the stabilization of the cellular environment via cell-cell and cell matrix junctions as well as ECM protein secretion over time. In other words, shape provides a measure of a stem cell's adaptation to prevailing mechanical stimuli prior to fate determination. Furthermore, cell shape evolves during tissue genesis, both during development as well as during processes of postnatal healing. In this way, D'Arcy Thompson's molding of form by physical forces becomes a modern molding of fate by physical forces that persist and stabilize form to a steady-state over time, i.e., stem cell lineage commitment is a manifestation of mechanoadaptation over time, reaching an equilibrium or steady state.

THE CELL ITSELF AS SENSOR, TRANSDUCER, ACTUATOR

Storage of information in cellular machinery

The cell itself, from its undifferentiated to its terminally differentiated state, is brainless yet retains functional memory in its structure, which becomes entrained or conditioned as structural connections are reinforced through repeated activity, ultimately stabilizing or entraining functional capacities.¹ For example, whether a cell is in a motile or adherent state is determined by the relative equilibrium or dynamic steady-state of cell-cell junctions and cell-matrix junctions, where the need to seek new food sources or literally a new environment (in the case of low oxygen tension in tumors), the motility machinery is upregulated genetically.

The dynamic stability and agility of the cell itself depends on the relative stability of its own structure as well as its capacity to adapt at time scales relevant for, e.g. cell division, motility, extracellular matrix secretion, etc. The cytoskeleton itself, which to a large extent defines cell morphology and cell state, stores information with regard to the state of the cell and its function in time. A critical aspect of understanding these relationships is in what cellular structures the information can be stored and the inherent time scales of information transfer as well as its effect on cellular function.

Tissue as a repository of cellular history affording memory as well as energy sources

The tissue itself reflects the history of the cells that inhabit and secrete the extracellular matrix proteins comprising the tissue over time. One may argue that this presents a circular argument, but when one considers non-motile cells, the cells themselves are the dominant modulators of their own environment. In contrast, motile cells ingress into new environments and thus must exhibit greater capacity to adapt.³ Created by the cell itself, the extracellular matrix comprising the structural components of tissue reflects the long term memory of the cellular experience.

Hence, cellular niches are created by non-motile cells, suggesting that the most specialized niche stabilizes phenotype for terminally differentiated cells. In contrast, the potential of stem cells is maximal (least specialized but with greatest potential to specialize along multiple lineages) in quiescent niches, like seed pods. Of particular note, stem cells appear to reside predominantly in ‘border zones’, such as the bone marrow, intestine and skin.^42-47 These tissue borders are defined by the presence of basement membranes (BM) which separate cells from the surrounding ECM. Furthermore, the BM has been reported to create niches of stem cells between the basal lamina and various tissues, including skeletal muscle,⁴⁸ epithelium,^42,49-52 hair follicles,^50,53,54 peripheral nerves,⁵⁵ blood vessels,⁵⁶ bone^57-61 and teeth.⁶² A recent discovery that stem cells are generated during mesenchymal to epithelial transitions provides further support for this hypothesis.^63-65

Similarly, mechanical activation of a previously quiescent niche appears to drive specialization of the cells through mechanoadaptation and eventually differentiation.² The length and time scales of cellular processes play an integral role encompassing cellular memory, enabling the cell to adapt to best survive follow on experiences whose probability depends on previous experiences.

In this way, complexity emerges from the most basic building blocks of life. Genes code for proteins comprising 23 amino acids. Proteins make up structural and energy storing elements, which in turn allow for complex spatiotemporal behavior. Behaviors of increasing complexity require structures of analogous complexity and complex behavior incorporates temporal components including behavioral stability, or resistance to change, versus dynamic “ability,” or capacity to adapt rapidly to sudden changes in environment. Recent studies show that the most basic property of cells, e.g., cell size, is modulated by the same physical laws as tissue and body size, namely the laws of gravity.^66,67 For example, oocytes of the frog Xenopus laevis grow bigger than 1 mm, with a nucleus of greater than 450 μm, in diameter. Counterintuitive to the field's most recent understanding of cell mechanics, a scaffold of elastic F-actin was found to stabilize “these large nuclei against gravitational forces.”⁶⁴ With the growth of the field in the future, it is expected that the mechanical constituents and properties of different cells and their parts, as well as their effects on tissue and organ physiology, will become more widely understood and quantitatively characterized.

Energy and work of cellular processes

One approach to assessing the probability that a cell will take one of the many possible paths depicted schematically in Waddington's landscape would be to do a calculation of the net energy cost of each path, where the probability of a given path increases with decreasing energy cost or respectively decreases with increasing energy cost. In this context, quiescence would reflect the relative tendency to maintain the status quo in a state minimizing e.g. metabolic cost. Currently, the specific energetic costs of cellular processes such as proliferation, migration and differentiation are unknown. Similarly, the energy cost of changing cellular structure or patterns of gene transcription are poorly understood.

Cutting edge methods to measure cell and subcellular deformation precisely under specific applied forces or stresses allow for an assessment of the energy landscape as lineage commitment occurs. For instance, controlling and measuring the forces applied at cellular boundaries while measuring the deformation of the cell allows for an in situ mechanical test of living cells while assessing changes e.g., in transcription of genes coding for structural proteins or genetic hallmarks of key differentiation milestones.²¹ Such experimental approaches provide unprecedented experimental parameters that can be used to assess the work of cellular processes over time as well as the transfer (exchange) of energy and mass between the cell and its environment. Current studies show clear correlations between mechanical cues, cellular deformation and genetic markers of lineage commitment and will be key in defining libraries of mechanical cues conducive to driving cell shape, mechanoadaptation and ensuing fate (Fig. 3). In parallel with mathematical models that account for energy and/or power, it is possible to assess the work and/or energy transfer among components of the system and to probe causal relationships^21,66 (see Figs. 3, 4 for examples with stem cells and cardiocytes). A major advantage of mathematical or computational models is that hypotheses can be tested virtually and used to prioritize which experiments will yield biggest insights into mechanisms underpinning cellular mechanotransduction and adaptation.⁶⁹

FIGURE 4. Experimental approach of Zile et al. to measure constitutive properties, including elastic properties (A-C) and viscous damping (D-F) of isolated cardiac muscle cells, in relation to cytoskeletal adaptation, i.e. microtubule and myofilament activation, in healthy and pressure overload hypertrophied hearts. (A-C) First the authors investigate passive spring properties of the 2 population of cells, by minimizing myofilament activation through chemical treatment while applying force to the gel in which cells are embedded. (D) The energy expended is then compared with that returned (E) to estimate viscous damping (F). Work refers to the process of applying a force over a distance; energy is the ‘cost’ of doing the work. Power is the rate of doing work or the rate of using energy to effect the work. Figures used with permission.⁶⁸

PREDICTIVE, MULTISCALE MODELING OF STRUCTURE-FUNCTION RELATIONSHIPS

To probe structure-function relationships underlying the mechanoadaptation hypothesis, it will be important to predict the work or energy cost of adapting cellular volume, shape and architecture throughout the differentiation process as well as the exchange of energy and mass between the cell and its environment. Two key issues must be addressed when creating a mathematical model aiming to elucidate causal relationships between cell shape and fate. First, the interplay between mechanics and chemistry at the cell and subcellular length scale, referred to as mechanochemistry, requires consideration. Then, the multiscale adaptation of cellular structure to function in its prevailing, dynamic mechanical environment, referred to as mechanoadaptivity, should be accounted for.

Mechanics and biochemistry are tightly coupled in cell signaling and physiology. Cells experience forces either through their membrane, via the fluidic environment as drag forces or osmotic stress, and/or direct strain/stress induced by deformation or displacement of boundaries including the nuclear envelope and its interactions with structures within and outside of the nucleus as well as the transmembrane proteins and their interactions with the internal cytoskeleton as well as external cell-cell and cell-matrix adhesion complexes. In both mechanochemical as well as mechanoadaptive processes, mechanical forces have a direct mechanical effect, governed by balance and compatibility conditions, and an indirect effect, related to the mechanotransduction of the mechanical stimuli into chemical signals within the cell. A robust, quantitative characterization of cell behavior (structure-function relationships) should account explicitly for both the biochemistry of the cell as well as the cell's mechanical feedback.

Furthermore, when a system of forces is applied via the cell membrane or through the transmembrane proteins, a reorganization of the cytoskeleton takes place. Actin and tubulin polymerize and depolymerize, resulting in spatiotemporal evolution of the cytoskeleton. In addition to modulating cell shape, the evolution of the actin and tubulin network results in a heterogeneous distribution of stiffness in the cell body and membrane.^7,8,15,16,30 Furthermore, cytoskeletal reorganization induces large deformations (in both shape and volume) of the cell and the nucleus, which, in turn, change the cell's mechanical environment, triggering a new reorganization of the cytoskeleton. Eventually, the overall process equilibrates, achieving an optimal state for the prevailing, dynamic mechanical environment. In the process, phenotype is stabilized and the “cell's fate is sealed.”

Neither mechanochemistry nor mechanoadaptivity enters the realm of classical continuum mechanics, necessitating the development of innovative approaches. Using the concept of virtual power and encompassing classical models such as plasticity and viscoelasticity, DiCarlo and Quiligotti^70,71 developed an elegant and effective method to treat materials with evolving microstructure. DiCarlo and Quiligotti's method is a special application of the theory of generalized continua introduced by Germain in the 1970s^72,73 and further developed by Capriz and others in the late 1980s.⁷⁴

Applied to biological systems, the theory provides a mechanistic framework to describe non-mechanical phenomena, such as growth and remodeling, through their mechanical feedback. The theory, described briefly in the following sections, has since been further developed to describe the remodeling of bone tissue^75,76 as well as biological processes such as the growth of membranes and tissues (Fig. 5). It provides an innovative approach and lends itself well⁷⁷ to elucidate mechanisms underlying the stem cell's innate capacity to adapt to mechanical stimuli as well as the relation between the stem cell's mechanoadaptive behavior and its lineage commitment.

FIGURE 5. Application of the virtual power theory to growth and remodeling of bone and saccular aneurysms. Bone tissue microstructure (A) at different anatomical locations within the femoral head. Used with permission.^78-81 Modeling the remodeling process as a rotation of the material axes: (B) states of remodeling equilibrium, adapted with permission,⁸¹ (C) evolution of the microstructure to a remodeling equilibrium point for different initial conditions (top) and of the corresponding principal strain ellipsoid (bottom), adapted with permission.^80,81

General framework

To address the change or adaptation in physical structure and subsequent biological function, a mathematical approach is developed for an idealized model of the cell, where the cell is treated as a collection of infinitesimally small material elements (Fig. 6). Conceptually, the model predicts the power (energy expended over time) during a virtual mechanical test of the cell (compare to experimental tests depicted in Figs. 3, 4). The model can also be treated iteratively to predict mechanoadaptation of a cell over time.

FIGURE 6. Kinematic description of a growing and remodeling cell on a substrate, where the idealized cell can be represented as a collection of material elements (single element depicted as turquoise square). Besides the classical description based on the placement of a body point (x) and of its gradient (Grad x), the evolution of the material microstructure are described by the growth tensor (G) and remodeling variables (m,R). The elastic distorsion tensor is F = (Grad x) G⁻¹.

The theory is based on a smart description of a material point's evolution, a proper balance statement of the generalized forces triggering this evolution, and a suitable constitutive theory.

Generalized kinematics

Besides its visible deformation, the evolution of a single material element (depicted as turquoise square, Fig. 6) undergoing internal reorganization can be addressed in terms of growth, which relates to the development of residual stress/strain in the material point. It can also be addressed in terms of remodeling, which encompasses the evolution of the material element's microstructure. In this framework, the complete motion of the material element is provided by the time evolution of its position in space (x) and of other kinematic descriptors related to growth (G) and remodeling (m,R).

Growth accounts for the point-wise change of the stress-free shape of the material element isolated from its neighbors and is described by the growth tensor G. The elastic distortion tensor F = (Grad x) G⁻¹ measures the difference between the visible configuration of the material element (described by Grad x) and the relaxed configuration (described by G). Remodeling accounts for the point-wise change of the elastic properties of the material element, which can be parameterized by the vector m of the elastic moduli and by the rotation R of the principal axes of elasticity. In sum, the complete motion of an element is a multi-variable process described by the set M = (x,G,m,R). During deformation and adaptation, any of those descriptors can change. Then, the complete velocity realized along the motion is V = (υ,W,ν,V), where υ is the visible velocity, W is the growth velocity and the pair (ν,V) is the remodeling velocity.

Generalized balance

The concept of power plays a major role in this theory. As the cell deforms, the associated forces and displacements (in the bulk as well as on the boundaries) can be understood in terms of power (the time rate of work, cf. Fig. 4.), which is defined as force applied over a velocity, as a function of time. Forces triggering internal reorganization of the cell can be understood in the same way, i.e. in terms of power expended to make the cell grow and remodel. In this way, forces are not introduced directly but by the value of power they produce for a given class of motions. In some sense, the power is the “conceptual gauge” sampling the evolution of the body. Any “measurable” change in a body element, related to either its visible deformation (Grad x) or the evolution of its microstructure (G,m,R), has to be related to some power expended, where power is the energy expended with respect to time. Thus, for a growing and remodeling body, the class of velocities shall encompass not only the “visible” velocity but also growth and remodeling velocities.

In the framework of the virtual power principle, force balance equations of a deforming body are obtained by requiring the total power expended by the forces on a suitable class of test velocities to be identically zero. This principle needs to be generalized when dealing with a growing and remodeling body^69,70 in order to account for the richer kinematic description encoded in the complete velocity V. Let V* = (υ*,W*,ν*,V*) be the set of test velocities associated to V. In the virtual power expression, generalized forces associated to the generalized test velocity V* have to be introduced. Thus, the generalized virtual power principle (GVPP) requires the total power expended by the generalized forces on the generalized test velocities to be identically zero. Of particular note in this theory, the terms “force” and “velocity” are used irrespective of their vectorial or tensorial nature. For the example in Fig. 5, the GVPP reads:(1) ${\displaystyle \begin{array}{l}\Pi ({\text{V}}^{\text{*}})-{\displaystyle {\int }_P(s^{\text{in}}\cdot {\upsilon }^{*}+S^{\text{in}}\cdot \text{Grad}{\upsilon }^{*})}\\ -{\displaystyle \underset{P}{\int }{A}^{\text{in}}\cdot W^{*}}-{\displaystyle \underset{P}{\int }{\beta }^{\text{in}}\cdot v^{*}}-{\displaystyle {\int }_P{B}^{\text{in}}\cdot V^{*}}\\ {\displaystyle \times {\int }_P{s}^{\text{out}}\cdot {\upsilon }^{*}}+{\displaystyle {\int }_{\partial P}{s}_{\partial }^{\text{out}}\cdot {\upsilon }^{*}}+{\displaystyle {\int }_P{A}^{\text{out}}\cdot W^{*}}\\ +{\displaystyle {\int }_P{\beta }^{\text{out}}\cdot v^{*}}+{\displaystyle {\int }_P{B}^{\text{out}}\cdot V^{*}}=0\end{array}}$(1) where P is a part of the body and ∂P its boundary. The first and second lines represent the powers expended by the generalized inner and outer forces, respectively, to be called inner (Πⁱⁿ) and outer (Π^out) powers for short. In Eq. (1(1) ${\displaystyle \begin{array}{l}\Pi ({\text{V}}^{\text{*}})-{\displaystyle {\int }_P(s^{\text{in}}\cdot {\upsilon }^{*}+S^{\text{in}}\cdot \text{Grad}{\upsilon }^{*})}\\ -{\displaystyle \underset{P}{\int }{A}^{\text{in}}\cdot W^{*}}-{\displaystyle \underset{P}{\int }{\beta }^{\text{in}}\cdot v^{*}}-{\displaystyle {\int }_P{B}^{\text{in}}\cdot V^{*}}\\ {\displaystyle \times {\int }_P{s}^{\text{out}}\cdot {\upsilon }^{*}}+{\displaystyle {\int }_{\partial P}{s}_{\partial }^{\text{out}}\cdot {\upsilon }^{*}}+{\displaystyle {\int }_P{A}^{\text{out}}\cdot W^{*}}\\ +{\displaystyle {\int }_P{\beta }^{\text{out}}\cdot v^{*}}+{\displaystyle {\int }_P{B}^{\text{out}}\cdot V^{*}}=0\end{array}}$(1) ) we introduced the mechanical inner (sⁱⁿ,Sⁱⁿ) and outer (s^out, s_∂^out) forces, the growth-related inner (Aⁱⁿ) and outer (A^out) forces and the remodeling-related inner (βⁱⁿ,Bⁱⁿ) and outer (β^out,B^out) forces. Inner forces develop in the body because it deforms, grows and remodels and have to be constitutively related to the motion M. Outer forces are the actions exerted on the body by the surrounding world. In particular, A^out and (β^out,B^out) encompass the mechanical feedback of the biochemical stimuli to growth and remodeling, respectively.

Constitutive theory

The main phenomenological ingredient is the description of the strain energy of the body Ψ(M). Moreover, evolution of the material properties is required to satisfy the principles of mechanics and thermodynamics. Two principles play a major role in this theory. First, the frame invariance principle, requiring the inner power Πⁱⁿ expended on any complete rigid velocity to be zero. For instance, considering rigid translations and rotations, this leads to the classical requirements for sⁱⁿ to be 0 and Sⁱⁿ to be symmetric-valued. Additional requirements may be set on (Aⁱⁿ,βⁱⁿ,Bⁱⁿ) according to the specific description of the complete rigid velocity.^69,70 Second, the dissipation principle is enforced, requiring the power expended during the actual motion of the body to be greater than the time rate of the energy,(2) ${\displaystyle -{\Pi }^{\text{in}}(\text{V)}-\dot{\Psi }\ge 0}$(2) where a superposed dot means time differentiation.

Application to idealized cell mechanoadaptation

Using the general theory described above to investigate stem cell mechanoadaptation, stem cell adaptivity results from point-wise deformation and reorganization of the cell structure, initially simplified as the ‘cell body’ (Fig. 5). During cell adaptation, the body element deforms and its microstructure evolves due to i.a. cytoskeleton reorganization, cell-cell and cell-matrix adhesions. At the scale of the cell, microstructural changes show up as growth and remodeling.

As an initial test of feasibility, a highly idealized model was created making the following assumptions. First (A1), small deformations are considered; then, the relevant strain measure is the small strain tensor E, i.e., the symmetric part of the gradient of the displacement. Secondly (A2), the cell body is considered as a homogeneous, linearly elastic, isotropic material; then, the vector m of the elastic moduli reduces to the bulk and shear moduli (K,μ) and the rotational remodeling (R) becomes meaningless; moreover, no distinction is made between the different parts of the cell body (i.a. cytosol, nucleus, organelles). Thirdly (A3), growth is ‘arrested’ (G = I). Fourthly (A4), strain energy density is chosen as a quadratic form of the strain. Finally (A5), a linear dissipation is associated with the evolution of the elastic moduli. In view of (A1-5) and following the ideas set forth previously,^3,20,21 the cell turns out to be modeled as an elastic body whose elastic moduli evolve according to the following pointwise evolution law for K,(3) ${\displaystyle d_K\dot{K}={\beta }_K^{\text{out}}-{\beta }_K^E\text{, with }{\beta }_K^E=\frac{3}{2}{\Vert E_{\text{sph}}\Vert }^2\text{and }{E}_{\text{sph}}=\frac{1}{3}(trE)I.}$(3) and a similar one for μ. In Eq. (3(3) ${\displaystyle d_K\dot{K}={\beta }_K^{\text{out}}-{\beta }_K^E\text{, with }{\beta }_K^E=\frac{3}{2}{\Vert E_{\text{sph}}\Vert }^2\text{and }{E}_{\text{sph}}=\frac{1}{3}(trE)I.}$(3) ), the non-negative coefficient d_K represents the dissipation related to the evolution of K: the higher d_K, the slower K will evolve. The outer stimulus β_K^out provides the energy supply needed for K to evolve and represents the mechanical feedback of the biochemical stimuli triggering remodeling. Without this external stimulus (β_K^out = 0), K would decrease to 0 (since β_K^E ≥ 0). Positive and negative values of β_K^out should be interpreted as a pointwise energy supply or loss, respectively, resulting e.g. from imbalance of catabolic and anabolic reactions taking place within the cell body; globally, this can be seen as an energy flow into or from the cell, respectively. Therefore, the model predicts that the elastic properties of the cell would naturally deteriorate and that the cell needs energy input to counterbalance this process. By contrast, deterioration is faster when the cell is required to release part of its bioenergetic resources to the environment. It is worth noting that this simplified model already contains the germs of a biologically sound process, in particular the need for some biochemical supply to sustain life as metabolism per se is a basic biological requirement of living cells, even in quiescent states. Interestingly, no ad hoc information is plugged into the model to drive it toward this specific outcome. As the theory is developed further, it is anticipated that cells and their constituents will be fully representable in their full, realistic complexity, including finite deformation, nonlinear elastic and inelastic behaviors, anisotropy, and heterogeneity.

Considering the cell as a spherical body subjected to a hydrostatic pressure p, β_K^E = 1/2(p/K),⁴ and assuming a similar power law for the external stimulus, β_K^out = α/2 (p/K)ⁿ, the evolution of K can be studied with respect to the pair of parameters (α,n) where arbitrary values are used for the other modeling parameters (Fig. 7). The only key point concerns the ratio p/K₀, proportional to the initial spherical strain, which was set smaller than one according to assumption (A1). Starting from the initial value K₀, the bulk modulus K can either decrease, stay stationary (for α = 1 and n = 2) or increase according to the specific pair of parameters (α,n). From a biological point of view, this would mean that the elastic properties of the cell will either deteriorate, not change, or improve as the cell tries to adapt to the external pressure. Of particular note, in case of “positive” adaptation, K always drops to zero as time goes by, pointing out a stable adaptive behavior triggering the system to a long-term steady state. Again, despite its simplicity, the model proves to be able to catch the basic features of cell adaptivity.

FIGURE 7. Time evolution of the bulk modulus and of its time rate. According to the value of the pair (α,n) parameterizing the external stimulus, this latter can either succeed or fail in triggering cell adaptation to the superimposed pressure.

BRIDGING COMPUTATIONAL AND EXPERIMENTAL MODELS

As a first test of feasibility, the idealized single cell model predicts hallmarks of cell behavior observed experimentally. Moving forward, it will be important to define prospective experimental approaches that will provide experimentally measured parameters enabling prediction of higher order behaviors with increasing degrees of biological complexity. Ultimately, it will be important to test the working hypothesis that stem cell fate is an emergent property resulting from persistence of structure-function relationships due to stabilization of the cellular environment via cell-cell and cell matrix junctions as well as extracellular matrix (ECM) protein secretion. The advantage of the proposed paired computational and experimental approach is its generalizability while also addressing specific changes in structure and function related to specialization of phenotype.

A number of key studies should be carried out to test whether it is possible to drive stem cell fate prospectively through delivery of controlled mechanical cues to stem cells seeded in both very compliant and stiffer, 3D milieu mimicking in vivo conditions. Specifically, it will be important to map, in situ, not only the stresses to which the live stem cell is exposed, but also the strain on the cell in response to the stress, as well as cell and nucleus shape, akin to an in situ mechanical test of the entire stem cell. Studies on live cells in engineered tissue templates demonstrate the feasibility of this approach and correlations indicate reference libraries of cues conducive to achieving specific cell fates. In addition, it will be important to probe structure – function relationships such as emergent anisotropy of the actin and tubulin cytoskeleton as well as lineage commitment as it unfolds. The molecular tools are available and have been tested for this purpose; the key aspect is tying the emergence of cytoskeletal and multicellular architectures while assessing emergent phenotype.

Moving forward to test the robustness of the working hypothesis, it will be important to use different cohorts, seeded or proliferated to target density to achieve target developmental contexts as well as primary cells in situ (in the mesodermal core of E11.5 murine embryos) or trypsinized and seeded at or proliferated to density.^5-7,82,83 In addition, it will be important to address the effect of developmental state (seeding protocols as well as compliance of the milieu), as well as independent variables include controlling for “cell adaptation” (living cell vs. fixed cell vs. cooled cell), cytoskeletal adaptation (including experiments to turn down or off polymerization of specific cytoskeletal components), as well as uncoupling of cell and cytoskeletal adaptations, cell spreading, and driving fate through biochemical means (Table 1).

Table 1. Example experimental design to test the theory of stem cell mechanoadaptation. (A) Libraries of mechanical cues will be tested prospectively using independent variables including (1) developmental state, (2) cytoskeletal adaptation, (3) uncoupling of cytoskeletal adaptation from cell shape and volume changes, as well as (4) biochemical controls. (B) Outcome measures will be made on dependent variables including the magnitude, distribution and degree of polymerization of cytoskeletal constituents including, e.g., actin and tubulin, cell and nucleus shape and volume as well as phenotypic measures such as gene expression and ECM protein synthesis. (C) Mechanical properties of the stem cells such as cell and nucleus elastic and shear moduli can be calculated using paired experimental - computation methods such as those demonstrated previously by Song et al.^12,20,21

A: INDEPENDENT VARIABLES LIBRARIES OF MECHANICAL CUES DEVELOPMENTAL STATE cells cultured at target density in multilayer assembly proliferate to density vs. seed at target density vs. substrate compliance CYTOSKELETAL ADAPTATION inhibit polymerization of actin with cytochalasin-D prevent disassembly of F-actin (but not secretion of actin) with 1 μM of myosin light chain kinase, ML-7^82 inhibit polymerization of tubulin with vinca alkaloid/cholchicine^81 prevent disassembly of microtubules with taxane^83 UNCOUPLE CYTOSKELETAL ADAPTATION FROM SHAPE, VOLUME CHANGESBlock plasma membrane anchoring proteins like β-catenin using primary cells from conditional knock out mice or soluble factors^80,81 CONTROLS: DRIVE FATE BIOCHEMICALLYusing e.g. chondrogenic vs. osteogenic vs. adipogenic inductive medium

B: DEPENDENT VARIABLES (OUTCOME MEASURES) ACTIN CYTOSKELETON (x,y,z,t) TOTAL ACTIN (intensity: concentration) POLYMERIZED ACTIN: (intensity: concentration) TUBULIN CYTOSKELETON (x,y,z,t) TOTAL TUBULIN (intensity: concentration) POLYMERIZED TUBULIN: (intensity: concentration) CELL SHAPE & VOLUME NUCLEUS SHAPE & VOLUME GENE EXPRESSION: QPCR WESTERN BLOT: ECM milieu

C: CALCULATED VARIABLES CELL BULK MODULUS: calculate from cell deformation under known boundary displacements NUCLEUS BULK MODULUS: calculate from nucleus deformation under known boundary displacements

“Mapping the mechanome” proceeds by creating 3D interaction plots between variables for selected outcome measures and testing the degree to which we can drive fate using the same variables as well as our experimental platforms that enable control of the cell's local environment. The use of both primary as well as immortalized stem cells and terminally differentiated cells from mouse embryos and adult humans, will be key to provide not only endogenous and experimental controls to test shape-fate relationships but also a robustness test for the relationship between species and anatomical sources, developmental stages and contexts, as well as between immortalized and primary cells.

To our knowledge, these data would comprise the first reference library for the mechanome of stem cells from embryonic and mature tissues. Hence, the innovation of the experimental approach lies in elucidation of cell and subcell mechanoadaptation as living cells in developing tissue templates grow (cell proliferation) and specialize (cell differentiation) and as mature tissues heal. Thus, these experiments will not only validate our predictive model but also broaden the impact of the model to control stem cell using biochemical and biophysical signals. An understanding of the relationship between a cell's mechanochemical environment and its shape and fate will open new possibilities to control cellular structure and function, not only for engineering but also for scale-up in manufacturing of tissue templates.

DISCLOSURE OF POTENTIAL CONFLICTS OF INTEREST

No potential conflicts of interest were disclosed.

Notes

1. Remarkably, just one quarter century ago, cell biology textbooks regarded stem cell differentiation and lineage commitment as a genetically preprogrammed process, guided by known chemical pathways. Only one decade ago, the concept of medicine personalized to individual's genomes drove the field to develop gene therapies. Since that time, the tide has changed; epigenetics or “the study of inherited changes in phenotype (appearance) or gene expression caused by mechanisms other than changes in the underlying DNA sequence, hence the name epi- (Greek: επί- over, above) -genetics” has gained power in the biological community, and D'Arcy Thompson's ideas appear to have withstood the test of time.