Physics and living matter
Although Galileo Galilei laid down the fundamentals for a mathematical description of natural processes, a long time passed before living organisms became a focus in physics. Given the high-spirited concept from René Descartes, living organisms were considered subject to the same universal laws as all objects. However, this approach did not provide scientists with an apparatus for systematically describing the physics of living matter. Isaac Newton’s examination of physical reality was based on analysis of the forces that cause movement of objects. Although he developed a very efficient and fruitful mathematical apparatus, living organisms appeared as “black boxes” therein. Terms such as “trajectory” and “force” became highly problematic when attempting to describe such phenomena as the growth, assembly, and physical activities of complex biological systems. These problems became apparent during the 20th century with advances in thermodynamics, molecular physics, computer simulations, and highly sophisticated experimental and imaging methods.
From the perspective of physics, living cells are considered physical systems, i.e., material structures composed of atoms and molecules governed by the laws of physics. A living cell becomes material with certain rheological properties. Its molecular structure is described using terms from macromolecular chemistry, polymer physics, statistical mechanics, and thermodynamics. Most of the biological attributes for living matter are reduced or neglected on purpose, as they are typically undefined using these physical methods. However, even physicists must reflect the enormous complexity of biological systems. When considering a living cell as a “material”, we cannot ignore that the phenomena, such as the “material”, is able to move spontaneously, and cells can migrate through pores and avoid barriers while simultaneously changing their shape as well as their viscoelastic and rheological properties. Moreover, these physical objects of interest are growing and multiplying in number, and they can even undergo cell death (e.g., apoptosis). Therefore, physicists must be careful when interpreting results from highly idealized models. This is an issue especially when confronting models with real experiments, as living systems typically do not behave as passive objects during an experiment. They are able to actively respond to the experimental conditions by changing their mechanical properties.
2. Thermodynamics of living structures
By publishing his “Reflections on the Motive Power of Fire” (Carnot, 1824), Sadi Carnot provided not only a theory and description useful for machine and design engineers, but he also laid the foundations for the second law of thermodynamics. At first, this theory was unnoticed as a universal natural law that cannot be simply deduced from physics based on Newtonian mechanics. This led Rudolf Clausius to become one of the founders of a new discipline: thermodynamics. Thermodynamics was based on physical quantities that were unknown until then, i.e., internal energy (
Let us consider a sample material at a resting state without any unbalanced forces between the system and adjacent systems and in a state of thermodynamic equilibrium with its surroundings. The macroscopic conditions in such a system may be described using a set of working parameters
The relationship between free energy and the parameters A is extremely important when describing our sample materials. Obviously, it is a measure of the
Free energy appears to be a key quantity in characterizing the rheological properties of living cells and tissues. When applying the laws of physics to living systems, it is a great challenge to describe the relationship between free energy and the working and structural parameters of the living system. However, both
Semiflexible networks typical in rubber-like materials have
The validation of these theoretical models seems to be successful, especially with experimental data characterizing artificial biopolymer networks created in vitro and separate from other biopolymers (e.g., collagen network, vimentin filaments, and fibrillar actin microfilaments). However, the in vivo experiments performed in living cells remain a problem, even for simple cell cultures, where cells are separate and free from the context of the entire tissue. The main issue is that the cells, which are supposed to be the object of the experiment, are remodeling during the experiment, thus altering its conditions. This raises a fundamental paradox: how can we measure the mechanical behavior of living cells if they react to our measurement tools (Bao & Suresh, 2003)?
Another issue is that living structures, unlike artificial biopolymers, behave as open systems, far from thermodynamic equilibrium. The cells require a constant influx of energy and behave as dissipative systems. For the application of non-equilibrium thermodynamics as devised by Onsager and Prigogine for cell membrane transport, see e.g., Dickel (1984). Biopolymer networks in vivo are in a non-equilibrium state, which results in the presence of a permanent mechanical prestress, not caused by any external force. The elastic behavior of these networks is non-linear (Fig. 2). Due to prestress, the material is more stiff, which has been demonstrated in experiments on actin cytoskeletal microfilaments (Gardel et al., 2006). While the elastic module for in vitro actin networks reaches 0. 1 Pa, the moduli in living cells were 10,000 times higher. Although this difference is also caused by the presence of additional networks, it can be concluded that prestress increases stiffness in living cells by at least 10 times. Prestress is based in part on active forces generated by molecular motors (such as within the contractile apparatus of muscle cells, see Fig. 3). Other chemical energy-driven phenomena might take part in prestress, such as ion pumps regulating cell volume and intracellular osmotic pressure.
3. The living cell as a physical system
The cell is the basic functional and morphological unit in living organisms (Alberts et al., 2002). The typical cell size is 10-20 µm, and the shape can be spheroidal, ellipsoidal, fusiform, polyhedric, or squamous. Cells are either free units or are surrounded by other cells and an extracellular matrix within tissue. There are two types of cells: prokaryotic
(bacteria) and eukaryotic (containing the cell nucleus and other membrane-bound compartments). Because of the plasma membrane (all cells) and even the cell wall (bacteria, plants, fungi, algae), the cell is an autonomous thermodynamic unit. The plasma membrane serves as an interface between intracellular and extracellular space. However, it does not insulate the cell from the extracellular environment. Rather, it is a selectively permeable barrier that hosts membrane transport systems. It is also involved in cell adhesion, cell signaling, ion conductivity, anchoring the cytoskeleton, attaching the extracellular matrix, maintaining cell volume, and the electric membrane potential. Maintaining stable and constant conditions within the cell, i.e., cell homeostasis, entails multiple dynamic equilibrium adjustments and regulatory mechanisms, which require approximately 30-70% of the energy available to the cell.
The intracellular space is compartmentalized by a number of membrane structures, while the space among them is filled with a colloidal liquid, the cytoplasm. In addition to the organelles and various inclusions, the intracellular space is reinforced by a scaffold, the cytoskeleton. This provides the cell with structure and shape. Cytoskeletal elements interact with cellular organelles and membranes. The cytoskeleton is also involved in cell motility, intracellular transport, cell division, and other processes.
From the physics point of view, the cell can be regarded as a structure comprised of heterogeneous materials, which correspond to the cytoskeleton, nucleus and other compartments. These constituents are considered continuous bodies with various material characteristics. Each of these materials is a viscoelastic continuum with a high degree of anisotropy and non-linear behavior. This approach is used to perform computational simulations of experiments performed with real cells using variety of techniques (e.g., atomic force microscopy as well as twisting, compressing, and pulling the cells, see Fig. 4-5). Describing the cellular structures as a sum of mechanical continua is useful only in tasks with dimensions exceeding the level of intracellular changes (Vaziri & Gopinath, 2008). Models based on the interaction of multiple physical phases can simulate interactions between the cytosol and solid objects using constitutive laws that describe the flow of fluid through a porous medium, such as Darcy’s law (Rohan & Cimrman, 2010). Such multiphase models are applicable when analyzing experiments based on the interaction of liquid and solid phases, such as micropipette aspiration experiments (Fig. 4C, see also Mofrad & Kamm, 2006).
However, for a better understanding of the cell as a physical system, more structures must be employed in our models. Probably the most important is the cytoskeleton (Fig. 6, see also Boal, 2002). The three main types of cytoskeletal filaments present in eukaryotic cells are microfilaments, intermediate filaments, and microtubules (Alberts et al., 2002). In prokaryotic cells, several types of structural and functional homologs of these cytoskeletal filaments were found (Carballido-López & Errington, 2003).
The actin filaments are the thinnest in the cytoskeleton at only 7 nm in diameter and are referred to as microfilaments. They are composed of linear actin polymers and are capable of generating force by elongation and shrinking at the ends. They also act as tracks for the movement of myosin molecules that attach to the microfilament, sliding along them. Actin microfilaments can be found both in the cortical and the deep cytoplasm. They are involved in cell motility, phagocytosis, cell division, and the transmission of force between the extracellular and intracellular structures. Several actin-binding proteins (e.g., α-actinin) contribute to actin cross-linking, thus modulating its in vivo mechanical properties. Unlike artificial microfilaments, in vivo actin is stiffer from prestress.
The intermediate filaments, approximately 10 nm in diameter, are more stable than actin filaments. Like actin filaments, they help maintain the cell-shape by resisting tensional forces. Therefore, the cells are able to resist a tension of 102-105 Pa (Bao & Suresh, 2003). Intermediate filaments organize the internal structure of the cell and anchor the organelles.
Microtubules are rigid, hollow cylinders approximately 25 nm in diameter, and are polymers of the protein tubulin. They exhibit a highly dynamic behavior as they polymerize and depolymerize. They are commonly organized by the centrosome, play role in the intracellular transport of organelles and vesicles, and construct the cilia, flagella, and mitotic spindles. They resist compression, and thus, in tensegrity models (tensional integrity, Stamenovic, 2006), they are considered rigid bars (rods). These rigid microtubules are interconnected by prestressed, extensible microfilaments (cables) (Fig. 7). These models are used to explain cellular tension and complex interactions between stiffness and prestress, thus describing even viscoelastic properties (Canadas et al., 2002). Tensegrity models may be combined with continuum models. However, they are highly sensitive to the spatial arrangement of the cables and rods.
As demonstrated by the tensegrity models, whenever physics aims to describe cell mechanics, it tends to choose a remarkable phenomenon, which becomes the basis for physical analysis. The cell has been considered a either mixture of polymer networks, a prestressed construction created from rods and fibers, or a multiphase continuum. Each of these strategies can explain certain features of cell behavior and may be used for planning and describing various experiments. Still, there is a demand for more general models, often inspired by non-trivial physical approaches. An interesting model is based on the similarity between cell behavior and the rheology of colloidal materials.
4. Biological cells and soft glassy materials
Despite the obvious presence of cytoskeletal semiflexible networks within the cells, physical description of these networks does not explain their complex behavior. The in vivo networks are not passive. On the contrary, the cell is a living structure; the networks undergo constant remodeling, and the activity of the molecular motors change over time. Despite the molecular level of these adaptive processes, their outcome is global and even macroscopic, i.e., at the level of the entire cell or tissue. Therefore, the next task is to describe the physical global effects from these molecular processes.
Recent advances in the rheology of soft glassy materials are at least a partial response to this problem (Fabry et al., 2001; Bursac et al., 2005; Deng et al., 2006). An experimental description of the frequency dependence for the complex modulus of the cell G(f) via magnetic cytometry (Fig. 4B) showed that the complex modulus of the cell consists of an elastic modulus G´ and a loss of this modulus iG´´, i.e., G = G´+ iG´´. These measurements were transformed into traditional elastic and loss moduli using a geometric factor, α, that depends on cell shape and thickness. For frequency dependence of the complex modulus for a semiflexible polymer network (e.g., fibrillar in vitro actin), G is either independent from the frequency (at lower frequencies) or is related (for higher frequencies) as follows:
Equation (3) complies with the theory of entropic elasticity. At higher frequencies, the heat-driven bending fluctuations of the semiflexible fibers contribute to the complex modulus. This relationship (3) has also been measured for cell cytoskeletons in vivo. However, there should be a considerable contribution from prestress in vivo, generating higher stiffness in the cytoskeletal network. Because of prestress, the energetic component should prevail over the entropic contribution (represented by the ¾ index in Equation (3)). Another discrepancy was found at lower frequencies. Unlike in vitro networks, the real cytoskeleton has a frequency dependence even at lower frequencies:
where α varies between 0.1-0.3. Such behavior no longer complies with semiflexible polymer networks. Instead, it corresponds to special materials referred to as soft glassy materials (Sollich, 1998). These material represent various foams, emulsions, slurries and colloidal systems. These “glassy materials” share features of structural disorder and metastability. Local elements in this material are either too large, or they are trapped by their neighbors. Therefore, they cannot rapidly reach heat-induced equilibrium, and their rearrangement via thermal activation, referred to as aging, is rather slow. However, this rearrangement can be facilitated by external energy (Fig. 8). The behavior depends on mesoscopic elements, which are regions small enough for a macroscopic piece of material to contain them in a large number, thus providing a description of its behavior as an average over elements. They are also large enough such that deformations at the scale of a single element can be described using an elastic strain variable (Sollich, 1998).
The application of these experiments and conceptual tools to the cytoskeleton provides evidence for a non-equilibrium cytoskeletal state. Prestress corresponds to the “densities” of colloidal particles trapped within the glassy system, causing the metastability. The ATP (adenosine triphosphate)-driven molecular motors (e.g., myosin, dynein, kinesin) and cytoskeleton-associated proteins rearrange the network in vivo, thus modifying the local non-equilibria within the cytoskeletal configuration. As a result, the macroscopic behavior of the cytoskeleton is observed (Fabry et al., 2001). The value of α is the central characteristic for the entire process. When α=1, the material is a fluid, and when α=0, the material is solid. As soft glassy materials, the cells may be near the glass transition. Upon activating the contractile apparatus, α (Equation 4) decreases to zero, which corresponds to the glassy transition. In contrast, the loosening and relaxing the cytoskeleton leads to an increase in α. The “glassy dynamics” appear at frequencies of 102 Hz, which is the transition frequency between the characteristics described by Equation (3) and (4). In other words, describing the cell in terms of soft glassy rheology is useful for processes at a scale of approximately 0.01 s. However, this is exactly the time scale for many cellular events, such as cell motility, movement of kinocilia, and phagocytosis, among others. (Deng et al., 2006).
It is noteworthy that when applying soft glassy rheology to cell behavior, the physical approach is far from a “naive reductionism”. There are no simulations at the molecular level. Conversely, the molecular processes remain in the background, varying only as a typical response to the stimulating frequencies. However, the “global” parameter, α, appears to be more important, as it characterizes the condition of the cytoskeleton and its typical response. The individual molecular mechanisms likely contribute to changes in α, thus affecting the global cellular response. Even the definition and experimental identification of such a parameter must be appreciated.
5. Cells and “intelligent materials”
In multicellular organisms, the cells are arranged in tissues, which usually compose organs. The cells are interconnected with cell-to-cell junctions that involve both plasma membranes and the intercellular matrix. The extracellular matrix consists of fibers and a ground “amorphous” substance. The prevailing matrix fibers in vertebrates are the collagen, elastin, and reticular fibers. The ground substance is constructed mainly of glycoproteins and glycosaminoglycans. In some tissues, the cells prevail, while other types of tissue are formed mainly by the matrix. Quantitative analysis of the tissue requires either a combination of microscopy and image processing or, preferably, unbiased stereological methods (Howard & Reed, 2005). The tissue can be characterized using volume fractions of the constituents (Fig. 9A), a description of the degree of anisotropy of the cells, the spatial orientation of the matrix fibers, cell clustering, the fiber length density, the microstructure surface area (Fig. 9B), and the numerical density of the objects in three dimensions, among others. Volumes, surfaces, lengths, numbers, and other microstructural characteristics can be estimated using stereological methods (Howard & Reed, 2005), which apply various geometrical principles for fair sampling of histological sections (Gundersen & Jensen, 1987). Stereology is based on methods related to the three-dimensional measurements available on the sections. A typical example includes the point-interception method, which estimates the area by counting the intersections between a randomly placed test grid and the objects of interest. The total number of points intersecting with the objects of interest is then multiplied by the reference area corresponding to each point. This is the reason stereology is also known as “stochastic geometry”.
In biological tissues, the cells are interconnected with the matrix fibers. The most abundant are collagen fibers. In fact, the hierarchical organization of collagen is complicated. Bundles of collagen fibers are constructed from fibrils, which consist of microfibrils, made from protofibrils that are overlapping trihelices of tropocollagen. In small deformations, the collagen fibers have a wavy appearance, and, unlike elastin fibers, they almost never contribute to the stress-strain behavior (Fig. 10, region I).
With increasing deformation (Fig. 10, region II), the collagen fibers become stretched and contribute to the total elastic modulus of the tissue. Further deformation (Fig. 10, region III) is characterized by fully stretched collagen fibers, which lead to sample stiffening. When increased stress reaches the ultimate strength of the tissue (region IV), microcracks are initiated and propagate through the tissue (Fig. 11). According to the arrangement of the collagen fibers, different tissues have different extensibility under uniaxial loading. For example, the mesentery of the peritoneal cavity can be distented by 100-200%, while ureters, arteries and veins can extend by approximately 60%; in addition, the cardiac muscle and tendons can extend to 15% and 2-5% of their initial length, respectively (Viidik, 1979; Ahlfors et Billiar, 2007). Therefore, we must consider that biological tissues have a complex microstructure and a non-linear, non-Hookean stress-strain response. Even under physiological conditions, they are subjected to large deformations.
From the perspective of rheology, the tissues can be considered viscoelastic materials, and therefore, they have properties such as stress relaxation, creep, and hysteresis. These parameters are identified using mechanical experiments, while preserving experimental conditions close to the in vivo state. The viscous properties result from biopolymers and the presence of intracellular and extracellular fluid. The intracellular fluid corresponds to approximately 40% of the tissue weight, while the extracellular fluid composes approximately 20% of the tissue weight. Deformation of the tissue is accompanied by redistribution of the fluid; however, the fluid renders the tissues incompressible.
Materials that can adapt to their environment and those that, more modestly, can alter their properties as required by their users are referred to as "intelligent". This is observed, for example, in the muscle tissue, which can change the microscopic contractility of the cytoskeletal filaments, causing either macroscopic peristalsis of the intestine, blood vessel vasoconstriction or blood vessel vasodilatation.
Let us consider a possible physical description for these mechanisms, enabling the number of N cells to modulate the mechanical behavior of the entire tissue. The free energy of a cell F
As the cells are much softer than the matrix, the consequence of Equation (5) is that the geometry of a cell is adapted to minimize deformation of the matrix, i.e., the cells will be deformed rather than the matrix. The contribution of the free energy is small; therefore, should the deformation mainly involve the cells, the intracellular processes would not be able to change the free energy of the entire tissue, i.e., they would not be able to alter the mechanical properties of the entire tissue.
In addition, the shape of the cells should not be deformed without limits, i.e., the values for parameters Ac
While the free energy ΔF
the geometric and mechanical parameters are optimized such that even with incompressible balls, the material behaves as if the balls were compressible, i.e., deformation of the material relies on deformation of the balls only. Under the condition described by Equation (8), the local inner energy ΔF’=0 (Fig. 13). However, even a small change in either δ
Nevertheless, do the living cells really use this principle? Taking the interaction between the actin and myosin myofilaments as an example, it is obvious that the sliding of the myosin heads along the actin microfilaments corresponds to changes in the initial lengths of the springs. Using these intrinsic processes driven by chemical energy, the cell changes the geometric parameters δ
Modeling the mechanical behavior of living cells and tissues is one of the biggest challenges in biomechanics. The models should comprise the properties of the cytoskeleton, plasma membrane, cell-to-cell junctions, as well as fibrillar and amorphous extracellular matrix. The macroscopic mechanical parameters of tissues can be identified much easier than microscopic parameters; therefore, the models serve as translators between macroscopic and microscopic phenomena. We reviewed several current approaches used in various branches of biophysics to devise non-trivial models of biological tissues.
Beginning with thermodynamics, which led us to study the free-energy dependence of microscopic parameters for living tissues, we continued with special mechanical properties of living cells (Bao & Suresh, 2003) and the important role of prestress in the cytoskeleton. We mentioned the tensegrity models (Stamenovic, 2006) and the approach in which the biological cell is regarded as a soft glassy material (Sollich, 1998). Active adaptations of intracellular structures require a physical description of such cytoskeletal remodeling (Bursac et al., 2005). We then discussed the problems with description and experimental identification of properties in living tissues and used an example to outline a purely mechanical explanation for the way that smooth muscle cells efficiently control the mechanical behavior of the entire tissue. We discussed the current methods for quantitative microscopic analysis of biological samples using stereological methods (Howard & Reed, 2005). We also outlined the importance of simple physical models in understanding the important features of biological structure functions (e.g., Holeček & Moravcová, 2006). To enhance this review’s availability to a broad audience, a number of illustrations accompanied the text.
We conclude that the mechanical properties of biological cells and tissues depend on their underlying complex microstructure. At the tissue level of organization, computer simulations based on a stochastic interpretation of real morphology allow us to pursue the effects from stiffness and elasticity of both the cytoskeleton and extracellular matrix on the mechanical behavior of entire organs. Despite the limitations in our knowledge and methods, physical models of cell mechanics offer powerful tools for researching the structural principles governing living matter.
The study was supported by the Grant Agency of Czech Republic under Project No. 106/09/0734 and Ministry of Education, Youth and Sports of the Czech Republic under Project No. MSM4977751303