Modeling the mechanics of fibrous-porous scaffolds for skeletal muscle regeneration.

The scaffolds for skeletal muscle regeneration are designed to mimic the structure, stiffness, and strains applied to the muscle under physiologic conditions. The external strains are also used to stimulate myogenesis of the (stem) cells seeded on the scaffold. The time- and location-dependent mechanics inside the scaffold determine the microenvironment for the seeded cells. Here, fibrous-porous cylindrical scaffolds under the action of external cyclic strains are considered. The scaffold mechanics are described as two-phase (poroelastic) where the solid phase is associated with the fibers and the fluid phase is associated with the liquid-containing pores. In response to an applied cyclic strain, pressure oscillates and fluid moves radially toward and away from the axis of the scaffold. We compute the directions and magnitudes of the radial gradients of the poroelastic characteristics (solid-phase displacement, strain, and velocity; fluid-phase pressure and velocity; relative fluid-solid-phase velocity) determined by the boundary conditions and geometry of the scaffold. Several kinds of the external cyclic strain are analyzed and the resulting poroelastic functions are found. The poroelastic characteristics are obtained in closed form which is convenient for further consideration of myogenesis of the seeded cells and ultimately for the design of the scaffolds for skeletal muscle regeneration. Graphical abstract.

A Poroelastic Model of a Fibrous-Porous Tissue Engineering Scaffold.

Tissue engineering scaffolds are used in conjunction with stem cells for the treatment of various diseases. A number of factors provided by the scaffolds affect the differentiation of stem cells. Mechanical cues that are part of the natural cellular microenvironment can both accelerate the differentiation toward particular cell lineages or induce differentiation to an alternative cell fate. Among such factors, there are externally applied strains and mechanical (stiffness and relaxation time) properties of the extracellular matrix. Here, the mechanics of a fibrous-porous scaffold is studied by applying a coordinated modeling and experimental approach. A force relaxation experiment is used, and a poroelastic model associates the relaxation process with the fluid diffusion through the fibrous matrix. The model parameters, including the stiffness moduli in the directions along and across the fibers as well as fluid diffusion time, are estimated by fitting the experimental data. The time course of the applied force is then predicted for different rates of loading and scaffold porosities. The proposed approach can help in a reduction of the technological and experimental efforts to produce 3-D scaffolds for regenerative medicine as well as in a higher accuracy of the estimation of the local factors sensed by stem cells.

Biophysical Stimulation for Engineering Functional Skeletal Muscle.

Tissue engineering is a promising therapeutic strategy to regenerate skeletal muscle. However, ex vivo cultivation methods typically result in a low differentiation efficiency of stem cells as well as grafts that resemble the native tissues morphologically, but lack contractile function. The application of biomimetic tensile strain provides a potent stimulus for enhancing myogenic differentiation and engineering functional skeletal muscle grafts. We reviewed integrin-dependent mechanisms that potentially link mechanotransduction pathways to the upregulation of myogenic genes. Yet, gaps in our understanding make it challenging to use these pathways to theoretically determine optimal ex vivo strain regimens. A multitude of strain protocols have been applied to in vitro cultures for the cultivation of myogenic progenitors (adipose- and bone marrow-derived stem cells and satellite cells) and transformed murine myoblasts, C2C12s. Strain regimens are characterized by orientation, amplitude, and time-dependent factors (effective frequency, duration, and the rest period between successive strain cycles). Analysis of published data has identified possible minimum/maximum values for these parameters and suggests that uniaxial strains may be more potent than biaxial strains, possibly because they more closely mimic physiologic strain profiles. The application of these biophysical stimuli for engineering 3D skeletal muscle grafts is nontrivial and typically requires custom-designed bioreactors used in combination with biomaterial scaffolds. Consideration of the physical properties of these scaffolds is critical for effective transmission of the applied strains to encapsulated cells. Taken together, these studies demonstrate that biomimetic tensile strain generally results in improved myogenic outcomes in myogenic progenitors and differentiated myoblasts. However, for 3D systems, the optimization of the strain regimen may require the entire system including cells, biomaterials, and bioreactor, to be considered in tandem.

Modeling Stem Cell Myogenic Differentiation.

The process of stem cell myogenesis (transformation into skeletal muscle cells) includes several stages characterized by the expression of certain combinations of myogenic factors. The first part of this process is accompanied by cell division, while the second part is mainly associated with direct differentiation. The mechanical cues are known to enhance stem cell myogenesis, and the paper focuses on the stem cell differentiation under the condition of externally applied strain. The process of stem cell myogenic differentiation is interpreted as the interplay among transcription factors, targeted proteins and strain-generated signaling molecule, and it is described by a kinetic multi-stage model. The model parameters are optimally adjusted by using the available data from the experiment with adipose-derived stem cells subjected to the application of cyclic uniaxial strains of the magnitude of 10%. The modeling results predict the kinetics of the process of myogenic differentiation, including the number of cells in each stage of differentiation and the rates of differentiation from one stage to another for different strains from 4% to 16%. The developed model can help better understand the process of myogenic differentiation and the effects of mechanical cues on stem cell use in muscle therapies.

Stem Cell Fate Decision Making: Modeling Approaches.

Mathematical (computational) modeling approaches can be effective tools in providing insight into cell-fate decisions. In this article, several major approaches to the modeling of embryonic, hematopoietic, adipose-derived, cancer, and neural stem cell differentiation are discussed. First, the population dynamics approach is considered. The models described as bifurcating dynamical systems that result in bistability or periodic oscillations are then discussed. Also, spatiotemporal models of cell differentiation, including continuum and discrete (agent- and rule-based) approaches, are reviewed. Further, the effects of the mechanical factors are discussed, including the convergence of the differentiation and mechanotransducton pathways and computational analysis of the extracellular matrix (surrounding tissue). Finally, the stochastic models that take into account the molecular noise of internal and external origins are reviewed. The effectiveness of the modeling in the creation of the improved differentiation platforms, elucidation of various pathological conditions, and analysis of treatment regiments has been demonstrated.

A Modeling Insight into Adipose-Derived Stem Cell Myogenesis.

Adipose-derived stem cells (ASCs) are clinically important in regenerative medicine as they are relatively easy to obtain, are characterized by low morbidity, and can differentiate into myogenic progenitor cells. Although studies have elucidated the principal markers, PAX7, Desmin, MyoD, and MHC, the underlying mechanisms are not completely understood. This motivates the application of computational methods to facilitate greater understanding of such processes. In the following, we present a multi-stage kinetic model comprising a system of ordinary differential equations (ODEs). We sought to model ASC differentiation using data from a static culture, where no strain is applied, and a dynamic culture, where 10% strain is applied. The coefficients of the equations have been modulated by those experimental data points. To correctly represent the trajectories, various switches and a feedback factor based on total cell number have been introduced to better represent the biology of ASC differentiation. Furthermore, the model has then been applied to predict ASC fate for strains different from those used in the experimental conditions and for times longer than the duration of the experiment. Analysis of the results reveals unique characteristics of ASC myogenesis under dynamic conditions of the applied strain.

The potential and electric field in the cochlear outer hair cell membrane.

Outer hair cell electromechanics, critically important to mammalian active hearing, is driven by the cell membrane potential. The membrane protein prestin is a crucial component of the active outer hair cell's motor. The focus of the paper is the analysis of the local membrane potential and electric field resulting from the interaction of electric charges involved. Here the relevant charges are the ions inside and outside the cell, lipid bilayer charges, and prestin-associated charges (mobile-transferred by the protein under the action of the applied field, and stationary-relatively unmoved by the field). The electric potentials across and along the membrane are computed for the case of an applied DC-field. The local amplitudes and phases of the potential under different frequencies are analyzed for the case of a DC + AC-field. We found that the effect of the system of charges alters the electric potential and internal field, which deviate significantly from their traditional linear and constant distributions. Under DC + AC conditions, the strong frequency dependence of the prestin mobile charge has a relatively small effect on the amplitude and phase of the resulting potential. The obtained results can help in a better understanding and experimental verification of the mechanism of prestin performance.

The local forces acting on the mechanotransduction channel in hair cell stereocilia.

In hair cells, mechanotransduction channels are located in the membrane of stereocilia tips, where the base of the tip link is attached. The tip-link force determines the system of other forces in the immediate channel environment, which change the channel open probability. This system of forces includes components that are out of plane and in plane relative to the membrane; the magnitude and direction of these components depend on the channel environment and arrangement. Using a computational model, we obtained the major forces involved as functions of the force applied via the tip link at the center of the membrane. We simulated factors related to channels and the membrane, including finite-sized channels located centrally or acentrally, stiffness of the hypothesized channel-cytoskeleton tether, and bending modulus of the membrane. Membrane forces are perpendicular to the directions of the principal curvatures of the deformed membrane. Our approach allows for a fine vectorial picture of the local forces gating the channel; membrane forces change with the membrane curvature and are themselves sufficient to affect the open probability of the channel.

Effects of plasma membrane cholesterol level and cytoskeleton F-actin on cell protrusion mechanics.

Protrusions are deformations that form at the surface of living cells during biological activities such as cell migration. Using combined optical tweezers and fluorescent microscopy, we quantified the mechanical properties of protrusions in adherent human embryonic kidney cells in response to application of an external force at the cell surface. The mechanical properties of protrusions were analyzed by obtaining the associated force-length plots during protrusion formation, and force relaxation at constant length. Protrusion mechanics were interpretable by a standard linear solid (Kelvin) model, consisting of two stiffness parameters, k0 and k1 (with k0>k1), and a viscous coefficient. While both stiffness parameters contribute to the time-dependant mechanical behavior of the protrusions, k0 and k1 in particular dominated the early and late stages of the protrusion formation and elongation process, respectively. Lowering the membrane cholesterol content by 25% increased the k0 stiffness by 74%, and shortened the protrusion length by almost half. Enhancement of membrane cholesterol content by nearly two-fold increased the protrusion length by 30%, and decreased the k0 stiffness by nearly two-and-half-fold as compared with control cells. Cytoskeleton integrity was found to make a major contribution to protrusion mechanics as evidenced by the effects of F-actin disruption on the resulting mechanical parameters. Viscoelastic behavior of protrusions was further characterized by hysteresis and force relaxation after formation. The results of this study elucidate the coordination of plasma membrane composition and cytoskeleton during protrusion formation.

Modeling electrically active viscoelastic membranes.

The membrane protein prestin is native to the cochlear outer hair cell that is crucial to the ear's amplification and frequency selectivity throughout the whole acoustic frequency range. The outer hair cell exhibits interrelated dimensional changes, force generation, and electric charge transfer. Cells transfected with prestin acquire unique active properties similar to those in the native cell that have also been useful in understanding the process. Here we propose a model describing the major electromechanical features of such active membranes. The model derived from thermodynamic principles is in the form of integral relationships between the history of voltage and membrane resultants as independent variables and the charge density and strains as dependent variables. The proposed model is applied to the analysis of an active force produced by the outer hair cell in response to a harmonic electric field. Our analysis reveals the mechanism of the outer hair cell active (isometric) force having an almost constant amplitude and phase up to 80 kHz. We found that the frequency-invariance of the force is a result of interplay between the electrical filtering associated with prestin and power law viscoelasticity of the surrounding membrane. Paradoxically, the membrane viscoelasticity boosts the force balancing the electrical filtering effect. We also consider various modes of electromechanical coupling in membrane with prestin associated with mechanical perturbations in the cell. We consider pressure or strains applied step-wise or at a constant rate and compute the time course of the resulting electric charge. The results obtained here are important for the analysis of electromechanical properties of membranes, cells, and biological materials as well as for a better understanding of the mechanism of hearing and the role of the protein prestin in this mechanism.

Internal forces, tension and energy density in tethered cellular membranes.

We analyze tethered cellular membranes by considering the membrane resultants, tension and densities of two modes of energy, bending and adhesion. These characteristics are determined based on a computational (finite-difference) analysis of membrane shape. We analyze the relative contribution and distribution of the membrane characteristics in four typical zones of the membrane surface. Using an axisymmetric model, we found that the meridional and circumferential components of the resultant are different near the tether body and they converge to the value of membrane tension farther from the tether. At the beginning of the area of membrane detachment from the cytoskeleton, the density of bending energy is on the same order of magnitude as membrane tension (resultant). Away from the tether, the bending energy density quickly decreases and becomes of the same order as that of the adhesion energy in the membrane-cytoskeleton attachment area. In that area, both modes of energy are significantly smaller than the membrane tension. We also consider the effect of the membrane bending modulus on the distribution of the membrane characteristics. An increase in the bending modulus results in changing the length and position on the membrane surface of zone 1 characterized by significant evolution of the resultant components. It also results in shortening zone 2 that covers the rest of the area of membrane detachment. The obtained results can help in a better interpretation of the measurements of membrane mechanical properties as well as in analyses of proteins and channels in curved membranes.

Stereocilia membrane deformation: implications for the gating spring and mechanotransduction channel.

In hair cells, although mechanotransduction channels have been localized to tips of shorter stereocilia of the mechanically sensitive hair bundle, little is known about how force is transmitted to the channel. Here, we use a biophysical model of the membrane-channel complex to analyze the nature of the gating spring compliance and channel arrangement. We use a triangulated surface model and Monte Carlo simulation to compute the deformation of the membrane under the action of tip link force. We show that depending on the gating spring stiffness, the compliant component of the gating spring arises from either the membrane alone or a combination of the membrane and a tether that connects the channel to the actin cytoskeleton. If a bundle is characterized by relatively soft gating springs, such as those of the bullfrog sacculus, the need for membrane reinforcement by channel tethering then depends on membrane parameters. With stiffer gating springs, such as those from rat outer hair cells, the channel must be tethered for all biophysically realistic parameters of the membrane. We compute the membrane forces (resultants), which depend on membrane tension, bending modulus, and curvature, and show that they can determine the fate of the channel.

Effect of membrane mechanics on charge transfer by the membrane protein prestin.

Prestin was found in the membrane of outer hair cells (OHCs) located in the cochlea of the mammalian inner ear. These cells convert changes in the membrane potential into dimensional changes and (if constrained) to an active electromechanical force. The OHCs provide the ear with the mechanism of amplification and frequency selectivity that is effective up to tens of kHz. Prestin is a crucial part of the motor complex driving OHCs. Other cells transfected with prestin acquire electromechanical properties similar to those in the native cell. While the mechanism of prestin has yet to be fully understood, the charge transfer is its critical component. Here we investigate the effect of the mechanics of the surrounding membrane on electric charge transfer by prestin. We simulate changes in the membrane mechanics via the corresponding changes in the free energy of the prestin system. The free energy gradient enters a Fokker-Planck equation that describes charge transfer in our model. We analyze the effects of changes in the membrane tension and membrane elastic moduli. In the case of OHC, we simulate changes in the longitudinal and/or circumferential stiffness of the cell's orthotropic composite membrane. In the case of cells transfected with prestin, we vary the membrane areal modulus. As a result, we show the effects of the membrane mechanics on the probabilistic characteristics of prestin-associated charge transfer for both stationary and high-frequency conditions. We compare our computational results with the available experimental data and find good agreement with the experiment.

Voltage-induced bending and electromechanical coupling in lipid bilayers.

The electrical properties of the cellular membrane are important for ion transport across cells and electrophysiology. Plasma membranes also resist bending and stretching, and mechanical properties of the membrane influence cell shape and forces in membrane tethers pulled from cells. There exists a coupling between the electrical and mechanical properties of the membrane. Previous work has shown that applied voltages can induce forces and movements in the lipid bilayer. We present a theory that computes membrane bending deformations and forces as the applied voltage is changed. We discover that electromechanical coupling in lipid bilayers depends on the voltage-dependent adsorption of ions into the region occupied by the phospholipid head groups. A simple model of counter-ion absorption is investigated. We show that electromechanical coupling can be measured using membrane tethers and we use our model to predict the membrane tether tension as a function of applied voltage. We also discuss how electromechanical coupling in membranes may influence transmembrane protein function.

Computational analysis of the tether-pulling experiment to probe plasma membrane-cytoskeleton interaction in cells.

Tethers are thin membrane tubes that can be formed when relatively small and localized forces are applied to cellular membranes and lipid bilayers. Tether pulling experiments have been used to better understand the fine membrane properties. These include the interaction between the plasma membrane and the underlying cytoskeleton, which is an important factor affecting membrane mechanics. We use a computational method aimed at the interpretation and design of tether pulling experiments in cells with a strong membrane-cytoskeleton attachment. In our model, we take into account the detailed information in the topology of bonds connecting the plasma membrane and the cytoskeleton. We compute the force-dependent piecewise membrane deflection and bending as well as modes of stored energy in three major regions of the system: body of the tether, membrane-cytoskeleton attachment zone, and the transition zone between the two. We apply our method to three cells: cochlear outer hair cells (OHCs), human embryonic kidney (HEK) cells, and Chinese hamster ovary (CHO) cells. OHCs have a special system of pillars connecting the membrane and the cytoskeleton, and HEK and CHO cells have the membrane-cytoskeleton adhesion arrangement via bonds (e.g., PIP2), which is common to many other cells. We also present a validation of our model by using experimental data on CHO and HEK cells. The proposed method can be an effective tool in the analyses of experiments to probe the properties of cellular membranes.

Voltage and frequency dependence of prestin-associated charge transfer.

Membrane protein prestin is a critical component of the motor complex that generates forces and dimensional changes in cells in response to changes in the cell membrane potential. In its native cochlear outer hair cell, prestin is crucial to the amplification and frequency selectivity of the mammalian ear up to frequencies of tens of kHz. Other cells transfected with prestin acquire voltage-dependent properties similar to those of the native cell. The protein performance is critically dependent on chloride ions, and intrinsic protein charges also play a role. We propose an electro-diffusion model to reveal the frequency and voltage dependence of electric charge transfer by prestin. The movement of the combined charge (i.e., anion and protein charges) across the membrane is described with a Fokker-Planck equation coupled to a kinetic equation that describes the binding of chloride ions to prestin. We found a voltage- and frequency-dependent phase shift between the transferred charge and the applied electric field that determines capacitive and resistive components of the transferred charge. The phase shift monotonically decreases from zero to -90 degrees as a function of frequency. The capacitive component as a function of voltage is bell-shaped, and decreases with frequency. The resistive component is bell-shaped for both voltage and frequency. The capacitive and resistive components are similar to experimental measurements of charge transfer at high frequencies. The revealed nature of the transferred charge can help reconcile the high-frequency electrical and mechanical observations associated with prestin, and it is important for further analysis of the structure and function of this protein.

Kinetic and mechanical analysis of live tube morphogenesis.

Ribbon is a nuclear Broad Tramtrack Bric-a-brac (BTB) -domain protein required for morphogenesis of the salivary gland and trachea. We recently showed that ribbon mutants exhibit decreased Crumbs and Rab11-coincident apical vesicles and increased apical Moesin activity and microvillar structure during tube elongation. To learn how these molecular and morphological changes affect the dynamics of tubulogenesis, we optimized an advanced two-photon microscope to enable high-resolution live imaging of the salivary gland and trachea. Live imaging revealed that ribbon mutant tissues exhibit slowed and incomplete lumenal morphogenesis, consistent with previously described apical defects. Because Moesin activity correlates with cortical stiffness, we hypothesize that ribbon mutants suffer from increased apical stiffness during morphogenesis. We develop this hypothesis through mechanical analysis, using the advantages of live imaging to construct computational elastic and analytical viscoelastic models of tube elongation, which suggest that ribbon mutant tubes exhibit three- to fivefold increased apical stiffness and twofold increased effective apical viscosity.

Modeling the mechanics of tethers pulled from the cochlear outer hair cell membrane.

Cell membrane tethers are formed naturally (e.g., in leukocyte rolling) and experimentally to probe membrane properties. In cochlear outer hair cells, the plasma membrane is part of the trilayer lateral wall, where the membrane is attached to the cytoskeleton by a system of radial pillars. The mechanics of these cells is important to the sound amplification and frequency selectivity of the ear. We present a modeling study to simulate the membrane deflection, bending, and interaction with the cytoskeleton in the outer hair cell tether pulling experiment. In our analysis, three regions of the membrane are considered: the body of a cylindrical tether, the area where the membrane is attached and interacts with the cytoskeleton, and the transition region between the two. By using a computational method, we found the shape of the membrane in all three regions over a range of tether lengths and forces observed in experiments. We also analyze the effects of biophysical properties of the membrane, including the bending modulus and the forces of the membrane adhesion to the cytoskeleton. The model's results provide a better understanding of the mechanics of tethers pulled from cell membranes.

Outer hair cell active force generation in the cochlear environment.

Outer hair cells are critical to the amplification and frequency selectivity of the mammalian ear acting via a fine mechanism called the cochlear amplifier, which is especially effective in the high-frequency region of the cochlea. How this mechanism works under physiological conditions and how these cells overcome the viscous (mechanical) and electrical (membrane) filtering has yet to be fully understood. Outer hair cells are electromotile, and they are strategically located in the cochlea to generate an active force amplifying basilar membrane vibration. To investigate the mechanism of this cell's active force production under physiological conditions, a model that takes into account the mechanical, electrical, and mechanoelectrical properties of the cell wall (membrane) and cochlear environment is proposed. It is shown that, despite the mechanical and electrical filtering, the cell is capable of generating a frequency-tuned force with a maximal value of about 40 pN. It is also found that the force per unit basilar membrane displacement stays essentially the same (40 pNnm) for the entire linear range of the basilar membrane responses, including sound pressure levels close to hearing threshold. Our findings can provide a better understanding of the outer hair cell's role in the cochlear amplifier.

Effects of chlorpromazine on mechanical properties of the outer hair cell plasma membrane.

An optical tweezers system was used to characterize the effects of chlorpromazine (CPZ) on the mechanical properties of the mammalian outer hair cell (OHC) through the formation of plasma membrane tethers. Such tethers exhibited force relaxation when held at a constant length for several minutes. We used a second-order generalized Kelvin body to model tether-force behavior from which several mechanical parameters were then calculated including stiffness, viscosity-associated measures, and force relaxation time constants. The results of the analysis portray a two-part relaxation process characterized by significantly different rates of force decay, which we propose is due to the local reorganization of lipids within the tether and the flow of external lipid into the tether. We found that CPZ's effect was limited to the latter phenomenon since only the second phase of relaxation was significantly affected by the drug. This finding coupled with an observed large reduction in overall tether forces implies a common basis for the drug's effects, the plasma membrane-cytoskeleton interaction. The CPZ-induced changes in tether viscoelastic behavior suggest that alterations in the mechanical properties of the OHC lateral wall could play a role in the modulation of OHC electromotility by CPZ.