Energetics and forces in living cells

Cells are rarely subject to inertia. The Reynold’s number, Re, which describes the ratio of inertial to viscous forces in a system, is given by ρvl/μ, where ρ is density, v is velocity, l is a relevant length scale, and μ is viscosity. With each human cell typically about 10 µm across and moving at speeds no greater than 10 µm/s, cells experience a Re of 10⁻⁴, meaning that viscous forces on a cell are 10 000 times as great as those of inertia.³ 

In the absence of inertia and without a constant input of directed force, cellular motion would essentially stop, save for diffusion and fluid currents. But cells possess a sophisticated internal structural network termed the cytoskeleton that can both resist external load and produce mechanical forces of its own.

The cytoskeleton is made of several different proteins that assemble into ropes or tubes whose length ranges from about 100 nanometers to a few millimeters. The cytoskeleton’s stiff microtubules are thought to withstand compression and provide roadways along which other proteins move inside the cell. Loose, ropey strands known as intermediate filaments provide the cell with mechanical toughness and resist stretching. The protein actin assembles into thin strands that act like temporary struts that rapidly assemble and then disappear. And the component known as myosin organizes itself into protein assemblies that pull on actin.

All of those cytoskeletal components are essential, but actin and myosin are particularly interesting because they allow the cell to move, change shape, and exert forces on its surroundings. Together, the different components form a cross-linked mesh that is anchored to the cell’s nucleus, membrane, organelles, and surroundings.

Although the details of how the cytoskeleton functions are complex, one way to think about its physical properties is in terms of a continuum, Kelvin–Voigt model—also known as the Voigt material—which lumps all of the biological complexity into a spring and a dashpot, shown in figure 1. That model captures the property that cells flow and deform on the scale of minutes and hours but are solid-like on the scale of seconds or shorter. It also illustrates how important both the magnitude and the loading rate of a force can be for dictating the cellular response. Our sense of touch provides an elegant, though still debated, example of the principle. Fingertips contain nerves that are optimized to sense vibrations at about 200 Hz, which allows one to pick up subtle features such as roughness by running a finger across a surface. In that context, the loading rate, rather than the load itself, is the important stimulus.

Recent work has modeled the cell’s cytoplasm as a poroelastic material.⁴ In that way of thinking, the cytoplasm is a viscous goo flowing through a cytoskeletal network of pores. Certain parameters set the time scale for the flow: the size ξ of hydraulic pores, through which liquids pass; an elastic modulus E, which incorporates elasticity of the cytoskeleton; and the effective viscosity μ, which describes the slow movement of embedded cytoplasmic elements. Those concepts come together in a poroelastic diffusion constant D_p ~ Eξ²/μ, according to which the apparent diffusive motion of water depends not only on the viscosity of the flow but also on the stiffness and structure of the cytoskeleton.

When and where the cytoskeleton has an important effect on diffusive transport is set by a poroelastic Péclet number Pe, the ratio of the advection to diffusion, or vl/D_p, where v and l are characteristic velocity and length scales, respectively.⁴ If Pe ≫ 1, the pore size and elastic properties of the cytoskeleton dominate the movement of water through the cell. In a muscle cell, for instance, D_p is likely about 50 µm²/s, l about 10 µm, and maximal contraction velocity v about 50 µm/s, which yields a Pe of about 10. That large a value suggests that cytoplasmic flow could be a source of interior, dissipative load in the muscle. The contraction of the muscle plausibly forces water through the relatively tiny holes in the cytoskeleton, which would consume some of the energy the muscle could otherwise produce. Whether that is, in fact, an important source of energetic inefficiency in the heart (or other muscles) has, so far as we’re aware, not been addressed.

The poroelastic view of the cell also suggests that a cell can potentially move simply by controlling the flow of water across its membrane. Recent experimental evidence suggests as much: Last year, cancer cells were found to crawl through thin channels by taking up water in the front and then pushing it out behind them.⁵ 

In short, a surprising amount about cells can be learned simply by thinking about them as bags of viscous liquid (the cytoplasm) flowing in an elastic polymer network (the cytoskeleton). But more can be learned by shrinking the perspective several orders of magnitude—from the micron scale of single cells to the nanometer scale of single proteins inside them.

The world of proteins is small and noisy. Typically exerting piconewtons of force, a protein will perform just 10⁻²¹ N·m, or 6.2 meV, of work by moving 1 nm against 1 pN of resisting load. While moving through the cell, a protein also experiences Brownian motion because of its own thermal energy and collisions with surrounding water molecules. Boltzmann’s constant k_B times the absolute temperature T—about 4.1 pN·nm, or 25 meV, at room temperature—describes the thermal energy of each of the protein’s degrees of freedom.

That thermal noise sets a rough lower energy bound for processes at work in the cell. To see why, consider a protein that can occupy two conformations, one compact and one elongated as shown in the box below. The ratio of the two conformations follows a Boltzmann distribution exp(−ΔG/k_BT), where ΔG is the Gibbs free energy difference between the states. If ΔG = 1 k_BT, the ratio is 1/3, a meaningful difference in the abundance of one state over the other. If ΔG = 0.5 k_BT, the ratio is only 1/1.6, a smaller difference in populations that a cell may not easily discern. An applied force can dramatically tilt the balance between the conformations.

A rough upper bound to the energy scale relevant to single proteins is that provided by adenosine triphosphate (ATP), the energy currency of the cell. Our enzymes couple the release of energy stored in the bonds of ATP’s charged backbone to otherwise unfavorable processes, like motion and synthesis. Metabolic energy from our food is used to recharge the cell’s supply of ATP. Under the conditions in a typical cell, the available energy per ATP molecule is about 20 k_BT. The molecular machinery of the cell thus operates at energies between 1 and 20 k_BT. At first glance, it might seem impossible to do any useful work with devices that operate at just a few times the thermal background energy. However, as we’ll see, nature has evolved remarkable protein-based machines that do exactly that.

Proteins are nanomachines par excellence. Those precisely manufactured amino-acid sequences have an exact length, exact composition, and high information content—all properties that are unusual among manmade polymers. Those characteristics allow proteins to fold into well-defined three- dimensional shapes that perform countless and diverse duties in a cell. (For a brief tutorial on how proteins fold, see figure 2.) In this article, the focus will be on the class of proteins that interconvert chemical and mechanical cues; that class includes both motor proteins, which convert chemical energy into mechanical work, and mechanosensitive proteins, which convert mechanical deformation into chemical signals.

Although the details are still debated, a passable approximation is that the folded shape of a protein is dictated by the space-efficient packing of greasy, hydrophobic amino acids on the protein’s interior and by the exposure of charged, hydrophilic amino acids on its exterior. More directional interactions—for example, hydrogen bonds and electrostatic forces—additionally specify a protein’s conformation so that it corresponds to a local energy minimum in the complex conformational space that the protein inhabits. Note that proteins can inhabit more than one such energy minimum; transitions between energy minima, which correspond to changes in the protein’s shape, can convey information and, in some cases, couple the chemical energy of ATP hydrolysis to the production of useful work.

Shielding the average hydrophobic amino acid from water yields about 4 k_BT of energy, assuming perfect space filling; hydrogen bonds likewise contribute roughly 2 k_BT each. The loss of configurational entropy upon folding yields a penalty of 2 k_BT per amino acid. So the hydrogen bonds and configurational entropy cancel each other out and leave about 2–4 k_BT per amino acid for protein stabilization upon folding. For a typical protein with 100 amino acids in its hydrophobic core, that corresponds to as much as 400 k_BT. To appreciate whether that’s a lot or a little, recall that the relative probability for a protein to inhabit one of two possible states scales as exp(−ΔG/k_BT). In this case the probability of the protein becoming unfolded would be 1 in 5 × 10¹⁷³!

In reality, proteins are usually far less stable. A typical protein’s folding free energy is only 12 k_BT, or one-half an ATP’s worth of energy. Mammalian proteins unfold at temperatures of around 50 °C. In contrast, proteins from thermophilic bacteria can remain folded in boiling water, which suggests that life can evolve exceedingly stable proteins when doing so is necessary. Indeed, the difference in stabilities appears deliberate: Computer-aided redesign of proteins that operate around room temperature show that a few design changes can lead to dramatic increases in thermal stability.⁶ 

Why would a protein evolve for marginal stability? One hypothesis is that it does not matter how stable a protein is, as long as it lies above a necessary stability threshold. Once the minimum threshold is met, random evolutionary drift would then generally leave the stability of a protein near that threshold.⁷ Another possibility is that excessive stability prevents regular metabolic turnover: Proteins that can’t be broken down accumulate and form plaques like the amyloid aggregates of Alzheimer’s or Parkinson’s disease (see Physics Today, June 2013, page 16).

More fundamentally, the marginal stability of proteins may be intimately related to their role as chemical catalysts. Enzymes with complicated catalytic mechanisms adopt multiple conformations corresponding to distinct steps in their catalytic cycle.⁸ Consistent with that model, many thermophilic enzymes lose their catalytic activity at room temperature. If a protein inhabits an energy well that is too deep, it becomes stuck, uselessly frozen in place.

The coupling of protein dynamics to chemical catalysis is beautifully illustrated by the motor protein myosin, which works in muscle to convert chemical energy from ATP into force and directed motion.⁹ More specifically, it couples ATP hydrolysis to the swing of a lever arm by pulling on filaments of cellular actin. (See the article by Rob Phillips and Steve Quake, Physics Today, May 2006, page 38.) ATP turns out to be hydrolyzed while myosin is detached from the actin filament. Elegant experiments demonstrate that ATP hydrolysis is reversible within myosin’s active site, meaning that very little change in energy occurs while both adenosine diphosphate and a phosphate group, the products of the hydrolysis, remain bound at the active site. The stored energy is ultimately used in a power stroke—the pulling of the lever arm about 10 nm—that is triggered when myosin binds to actin and frees the phosphate, as outlined in figure 3.

To maintain the traction necessary to pull on a tensed filament, the myosin head must bind to actin tightly. The strong binding is broken only by the affinity of myosin for ATP; thus the strong interaction between myosin and actin is traded for an even stronger interaction between myosin and ATP. The net effect is that myosin efficiently couples chemical energy to directed force.

In addition to possessing molecular-scale machines such as myosin, various cells also contain numerous molecular force sensors that can detect mechanical stretch, substrate stiffness, fluid flow, vibration, compression, and, more broadly, heat and light.¹⁰ Because so many sensors stud a cell’s surface, understanding how each sensor functions is a major intellectual enterprise.

Perhaps the most-studied molecular mechanosensor complex consists of the protein assemblies that link the cell to the surrounding extracellular matrix. Those assemblies, termed focal adhesions, are breathtakingly intricate, composed of thousands of protein molecules and hundreds of different protein types. Despite the compositional complexity, the way in which focal-adhesion proteins sense force can be described by fairly simple physics.

Many force-sensing proteins are likely present in focal adhesions, but the best understood are the proteins talin and vinculin, which link adhesion proteins to the cellular actin network. Biological and biophysical measurements support the model shown in figure 4: Mechanical stretch unfolds domains within talin, to which vinculin can bind to reinforce the connection between the cytoskeleton and focal adhesion.¹¹ The stretch tilts the energy balance between the folded and unfolded states for specific portions of the talin protein (see the box). The unfolded domains recruit vinculin, which both binds to actin and recruits additional proteins. In essence, a mechanically driven physical transformation (unfolding) is converted into a protein-binding interaction, which acts as a form of information processing and storage.

A single myosin molecule exerts about 2 pN, and the exposure of each new vinculin binding site adds at least 5 nm to the stretched talin molecule. As a result, the force produced by just one myosin motor is sufficient to bias a vinculin binding site in talin toward its open state by 5 nm × 2 pN, or 2.4 k_BT. That results in a 10-fold shift in the equilibrium constant K toward the elongated, unfolded conformation. Four myosin molecules bound to an actin filament would lead to a 15 000-fold shift in K toward the elongated, unfolded state.

Force sensing also has mechanical consequences. Once it binds to talin, vinculin also changes conformation, such that it, too, can bind to actin, as pictured in figure 4. In that way, force sensing is automatically connected to reinforcement of a mechanical connection between the cell and its environment. Although the details still need to be worked out, it seems likely that this model and similar strategies allow the cell to automatically adjust the strength with which it adheres to its substrate, and likely to its neighbors, in response to variations in mechanical load.

The talin–vinculin example is just one particularly well-studied case of a myriad of protein sensors at work in living cells. In most cases, biophysicists know next to nothing about how a cell converts the effects of membrane bending, fluid flow, or the vibrations from sound or touch, say, into chemical information it can process. Fortunately, techniques for measuring molecular-scale forces in living cells and even intact animals are improving rapidly. The example of talin indicates that proteins that undergo nanometer changes in length can sense the piconewton-scale forces inside cells. As techniques improve, it will be fascinating to discover how the cell’s many other mechanosensors function at the molecular level.

Biophysicists are beginning to discover that many cellular functions rely on the dynamic assembly of hundreds of proteins and macromolecules into micron-scale structures. Focal adhesions constitute one such example. Others include the nucleolus—a micron-sized structure in the nucleus that assembles ribosomes—and the massive protein and RNA complexes that process messenger RNA. Signal transduction at the cell membrane is also likely to depend on large, roughly 100-nm protein and lipid structures that are still poorly understood (see the Quick Study by Ned Wingreen in Physics Today, September 2006, page 80).

Each of those highly complex examples is much smaller than a cell; the continuum models discussed earlier for the cytoskeleton and cytoplasm are clearly not appropriate. And yet a molecular-scale description that would be useful for single proteins is not practical for these assemblies because of their size and compositional heterogeneity. Understanding how they and other superstructures assemble and function is thus an open challenge for biologists and physicists.

In 1959 Richard Feynman laid out some of the major principles that physicists and engineers would need to consider when building machines on the nanometer scale,¹² a task that nature had already perfected over the past several billion years. Thanks to a half century of progress, scientists are now reasonably knowledgeable about how protein machines work and how to begin using biological principles as design rules for creating nanoscale machines of their own. However, our relatively sophisticated understanding of protein—and DNA and lipid—biophysics only highlights the gap in our understanding of how hundreds of molecular components come together to fulfill complex functions in the cell. While it may be that there is no longer “plenty of room at the bottom,”¹² there’s undoubtedly plenty of room in the middle.

Alex Dunn is an assistant professor of chemical engineering and Andrew Price is a doctoral student in biophysics, both at Stanford University in Stanford, California.