The present chapter is included in Part E with the other optical spectroscopy methods; however, the development of two-dimensional IR (2D-IR) spectroscopy is strongly based on two-dimensional NMR, and it is easier to understand after reading the relevant sections in Part J, which the reader is strongly encouraged to do first.

Historical review and introduction to biological problems

1950

O. Hann proposed coherent spectroscopy – the use of radiation fields with well-defined phase properties – to extract information about atoms and molecules. The ‘spin echo’ experiment in nuclear magnetic resonance was the first demonstration of the possibilities of coherent spectroscopy.

1957

R. P. Feynman, F. L. Vernon Jr. and R. W. Hellwarth published a landmark paper pointing out that if coherent light fields were ever created, it would be possible to use these same methods on optical transitions. The invention of the laser in 1960 was followed quickly by a demonstration of the ‘photon echo’ – the optical version of the ‘spin echo’.

1998

R. M. Hochstrasser and collaborators proposed 2D IR spectroscopy, in analogy with two-dimensional NMR, for the determination of time-evolving structures. The spins associated with the different nuclei in NMR are replaced in the IR experiments by a network of vibrational modes whose coupling can be used to determine molecular structure and dynamics. Structures of dipeptides, tripeptides and pentapeptides were determined by 2D IR spectroscopy. The most exciting aspect of 2D IR spectroscopy, however, is the combination of its sensitivity to structure with time resolution.

Historical review

In 1963 Richard Feynman traced to Pythagoras (c. 500bc) the first example, outside geometry, of the discovery of a numerical relationship in Nature. Pythagoras' discovery, which led to the foundation of a school of thought with mystic beliefs in the power of numbers, was that two strings under the same tension but of different lengths give a pleasant sound when plucked together if the ratio of their lengths is that of two small integers. We now say that a ratio of 1:2 corresponds to an octave, a ratio of 2:3 to a fifth and so on, which are all harmonic sounding chords. Feynman analysed this discovery in terms of three characteristics: its basis in experimental observation, the use of mathematics as a tool for understanding Nature, and its concern with aesthetics (the ‘pleasant’ quality of the sound). With easy hindsight (as he readily admits!) Feynman wrote that if Pythagoras had been more impressed by the first point on the importance of experimental observation the science of physics might have had an earlier start.

We discovered the existence and seek our biophysical understanding of macromolecules through experiment, and we use mathematical tools not only to set up experiments and analyse the results, but also for the description of the studied ‘objects’ themselves. The basis of aesthetics may still remain as mysterious as in Pythagoras' time, but there is no doubt that the ‘beauty’ of the DNA double helix and the satisfyingly elegant way in which it provided an explanation for how genetic information is stored and transmitted was an essential inspiration in the development of modern molecular biology.

Historical overview and introduction to biological applications

We recall from Chapter A1 that biological events occur on an immensely extended range of time scales – from the femtosecond of electronic rearrangements in the first step of vision, to the 109 years of evolution. Thermal energy is expressed as atomic fluctuations on the picosecond–nanosecond time scale that constitute the basis of molecular dynamics. These fluctuations are of particular interest in biophysics because they result from and reflect the forces that structure biological macromolecules and the atomic motions and molecular flexibility associated with biological activity.

Thermal energy propagates through solids in waves of atomic motion, such as the normal modes discussed in Chapters A3 and I1. We can estimate values for the frequencies, wavelengths and amplitudes of thermal excitation waves from an order of magnitude calculation, e.g. by considering the movement of a mass similar to the mass of an atom moving in a simple harmonic potential of energy equal to Boltzmann's constant multiplied by 300 K (ambient temperature). It turns out that the frequencies are of the order of 1012 s−1, while the wavelengths and amplitudes are on the ångström scale. We saw in Part E that the energy associated with thermal vibrations corresponds to the IR frequency range in optical spectroscopy. Similarly, neutron spectroscopy takes advantage of the fact that the energies of thermal neutron beams match vibrational energies in solids and liquids.

Historical review

1845

G. Stokes showed that the translational friction for a sphere is proportional to its radius, and to the viscosity of its surrounding solvent. In 1856 he demonstrated that for small angular velocity the rotation of the sphere may be characterised by a single parameter, which is proportional to the linear dimensions cubed.

1893

D. Edwardes calculated two frictional coefficients of the rotation for an ellipsoid of revolution: one for rotation around the axis of revolution and another for rotation around a direction normal to the first. In 1906 A. Einstein showed that rotation of the sphere in Stokes' approximation may be characterised by a single constant which has the dimensions of time. In 1928 R. Gans used the Edwardes frictional coefficients for an ellipsoid of revolution to calculate the ratios of the principal relaxation times to the relaxation time of a sphere of equal volume. In 1936 F. (Francis) Perrin presented equations that give the three rotational coefficients and three rotational relaxation times as functions of the dimensions of a three-axis ellipsoid. These equations could not be expressed in terms of elementary functions. In 1960 L. D. Favro showed that diffusion coefficients related to the rotational motion of a general particle involve five relaxation times; when two of the diffusion coefficients are equal the number of relaxation times is reduced to three. In 1977 E. Small and I. Isenberg solved Perrin's equations for the rotational diffusion of a general ellipsoid using a numerical integration procedure.

Historical review

1913

A. Dumansky proposed the use of ultracentrifugation to determine the dimensions of colloidal particles.

1923

T. Svedberg and J. B. Nichols constructed the first centrifuge with an optical system to follow particle behaviour in a centrifugal field. One year later, Svedberg noted the decrease in absorbance at the top of the cell during centrifugation of a haemoglobin solution.

1926

Svedberg made the first measurements of protein molecular weights (haemoglobin and ovalbumin) by sedimentation equilibrium and in 1927 he determined the molecular weight of haemoglobin using a combination of sedimentation and diffusion data. These pioneering studies led to the undeniable conclusion that proteins are truly macromolecules, made up of a large number of atoms linked by covalent bonds (Comment D4.1).

(Comment D4.1)

1929

O. Lamm deduced a general equation describing the behaviour of the moving boundary in the ultracentrifuge field. The exact solution of the equation is an infinite series of integrals, which can be computed only by numerical integration. In later work, the Lamm equation was solved analytically for specific limiting cases (H. Faxen, W. J. Archibald, H. Fujita).

1930s

Schlieren optical systems were designed by J. St. L. Philpot and H. Svenson, and independently by L. G. Longsworth; these allowed a representation of the concentration gradient (or, more precisely, the refractive index increment) as a function of distance in the centrifuge sample cell.

Historical review

1869

J. Tyndall performed the first experimental studies on light scattering from aerosols. He explained the blue colour of the sky by the presence of dust in the atmosphere.

1871

Lord Rayleigh presented the theory of scattering from assemblies of non-interacting particles that were sufficiently small compared with the wavelength of light. According to Rayleigh, scattering by a gas occurs because of the fluctuation of the molecules around a position of equilibrium. Rayleigh obtained the formulae that explained the blue colour of the sky as being due to preferential scattering of blue light in comparison with red light by molecules in atmosphere.

1906

L. Mandelshtam raised the question on the nature of scattered light once again. He pointed out that Raylegh's arguments do not fully explain the scattering phenomenon. According to Mandelshtam light scattering is also due to the random fluctuations of molecules near the position of equilibrium. As a result of random fluctuations of order λ3, where λ is the scattering wavelength, the number of macromolecules will vary with time. It follows from the Mandelshtam theory that the translational and rotational diffusion coefficients of macromolecules could be obtained from the spectrum of the scattered light.

1924

L. Mandelshtam described theoretically and experimentally so-called combination light scattering, which was later called Raman scattering. As early as 1926 he recognised that the translational diffusion coefficient of macromolecules can be obtained from the spectrum of the light they scatter.

EM in biology

Structural biology with EM

Electron microscopes can provide vivid details of biological macromolecules at the nanometre scale.

Recall from Chapter H1, an electron image is a two-dimensional projection of the object along the electron beam. A three-dimensional reconstruction of the object can be obtained by combining data measured at different angular projections. In single-particle reconstruction, a population of molecules is imaged. If these are identical (monodisperse) and present in sufficient quantities, the data from differently oriented molecules can be combined to produce an accurate three-dimensional reconstruction of the structure. The presence of symmetry is taken advantage of in helical reconstruction and two-dimensional crystallography. In tomography, the same object is imaged at different orientations. The different images are then combined together to recreate its structure.

Examples of electron cryo-microscopy reconstructions

In EM, the image is the result of the interaction of the incident electrons with the electrostatic distribution due to the atomic structure of the sample. The reconstruction of the structure results from the analysis of many images. Three-dimensional reconstructions at different resolutions depict the object at different levels of detail (Figure H2.1).

At atomic resolution, it is a good approximation to understand this structural density as being equivalent to the electron density obtained by X-ray diffraction, but with a larger contribution from hydrogen atoms in the electron diffraction case. It should furthermore be possible to distinguish the charge state of amino acids.

Introduction

All living things, despite their profound diversity, share a common architectural building block: the cell. Cells are the basic functional units of life, yet are themselves comprised of numerous components with distinct mechanical characteristics. To perform their various functions, cells undergo or control a host of intra-and extracellular events, many of which involve mechanical phenomena or that may be guided by the forces experienced by the cell. The subject of cell mechanics encompasses a wide range of essential cellular processes, ranging from macroscopic events like the maintenance of cell shape, cell motility, adhesion, and deformation to microscopic events such as how cells sense mechanical signals and transduce them into a cascade of biochemical signals ultimately leading to a host of biological responses. One goal of the study of cell mechanics is to describe and evaluate mechanical properties of cells and cellular structures and the mechanical interactions between cells and their environment.

The field of cell mechanics recently has undergone rapid development with particular attention to the rheology of the cytoskeleton and the reconstituted gels of some of the major cytoskeletal components – actin filaments, intermediate filaments, microtubules, and their cross-linking proteins – that collectively are responsible for the main structural properties and motilities of the cell. Another area of intense investigation is the mechanical interaction of the cell with its surroundings and how this interaction causes changes in cell morphology and biological signaling that ultimately lead to functional adaptation or pathological conditions.

A wide range of computational models exists for cytoskeletal mechanics, ranging from finite element-based continuum models for cell deformation to actin filamentbased models for cell motility.

ABSTRACT: Cells are highly complex structures whose physiology and biomechanical properties depend on the interactions among the varying concentrations of water, charged or uncharged macromolecules, ions, and other molecular components contained within the cytoplasm. To further investigate the mechanistic basis of the mechanical behaviors of cells, recent studies have developed models of single cells and cell–matrix interactions that use multiphasic constitutive laws to represent the interactions among solid, fluid, and in some cases, ionic phases of cells. The goals of such studies have been to characterize the relative contributions of different physical mechanisms responsible for empirically observed phenomena such as cell viscoelasticity or volume change under mechanical or osmotic loading, and to account for the coupling of mechanical, chemical, and electrical events within living cells. This chapter describes several two-phase (fluid-solid) or three-phase (fluid-solid-ion) models, originally developed for studying soft hydrated tissues, that have been extended to describe the biomechanical behavior of individual cells or cell–matrix interactions in various tissue systems. The application of such “biphasic” or “triphasic” continuum-based approaches can be combined with other structurally based models to study the interactions of the different constitutive phases in governing cell mechanical behavior.

Introduction

Cells of the human body are regularly subjected to a complex mechanical environment, consisting of temporally and spatially varying stresses, strains, fluid flow, osmotic pressure, and other biophysical factors. In many cases, the mechanical properties and the rheology of cells play a critical role in their ability to withstand mechanical loading while performing their physiologic functions. In other cases, mechanical factors serve as important signals that influence, and potentially regulate, cell phenotype in both health and disease.

ABSTRACT: Cell shape is an important determinant of cell function and it provides a regulatory mechanism to the cell. The idea that cell contractile stress may determine cell shape stability came with the model that depicts the cell as tensed membrane that surrounds viscous cytoplasm. Ingber has further advanced this idea of the stabilizing role of the contractile stress. However, he has argued that tensed intracellular cytoskeletal lattice, rather than the cortical membrane, confirms shape stability to adherent cells. Ingber introduced a special class of tensed reticulated structures, known as tensegrity architecture, as a model of the cytoskeleton. Tensegrity architecture belongs to a class of stress-supported structures, all of which require preexisting tensile stress (“prestress”) in their cable-like structural members, even before application of external loading, in order to maintain their structural integrity. Ordinary elastic materials such as rubber, polymers, or metals, by contrast, require no such prestress. Ahallmark property that stems from this feature is that structural rigidity (stiffness) of the matrix is nearly proportional to the level of the prestress that it supports. As distinct from other stress-supported structures falling within the class, in tensegrity architecture the prestress in the cable network is balanced by compression of internal elements that are called struts. According to Ingber's cellular tensegrity model, cytoskeletal prestress in generated by the cell contractile machinery and by mechanical distension of the cell. This prestress is carried mainly by the cytoskeletal actin network, and is balanced partly by compression of microtubules and partly by traction at the extracellular adhesions.

ABSTRACT: Using a novel method that was both quantitative and reproducible, Francis Crick and Arthur Hughes (Crick and Hughes, 1950) were the first to measure the mechanical properties inside single, living cells. They concluded their groundbreaking work with the words: “If we were compelled to suggest a model (of cell mechanics) we would propose Mother's Work Basket – a jumble of beads and buttons of all shapes and sizes, with pins and threads for good measure, all jostling about and held together by colloidal forces.”

Thanks to advances in biochemistry and biophysics, we can now name and to a large degree characterize many of the beads and buttons, pins, and threads. These are the scores of cytoskeletal proteins, motor proteins, and their regulatory molecules. But the traditional reductionist approach – to study one molecule at a time in isolation – has so far not led to a comprehensive understanding of how cells are able to perform such exceptionally complex mechanical feats as division, locomotion, contraction, spreading, or remodeling. The question then arises, even if all of the cytoskeletal and signaling molecules were known and fully characterized, would this information be sufficient to understand how the cell orchestrates complex and highly specific mechanical functions? Or put another way, do molecular events playing out at the nanometer scale necessarily add up in a straightforward manner to account for mechanical events at the micrometer scale?

We argue here that the answer to these questions may be ‘No.’ We present a point of view that does not rely on a detailed knowledge of specific molecular functions and interactions, but instead focuses attention on dynamics of the microstructural arrangements between cytoskeletal proteins.

ABSTRACT: Cells can be modeled as continuum media if the smallest operative length scale of interest is much larger than the distance over which cellular structure or properties may vary. Continuum description uses a coarse-graining approach that replaces the contributions of the cytoskeleton's discrete stress fibers to the local microscopic stress-strain relationship with averaged constitutive laws that apply at macroscopic scale. This in turn leads to continuous stress-strain relationships and deformation descriptions that are applicable to the whole cell or cellular compartments. Depending on the dynamic time scale of interest, such continuum description can be elastic or viscoelastic with appropriate complexity. This chapter presents the elastic and viscoelastic continuum multicompartment descriptions of the cell and shows a successful representation of such an approach by implementing finite element-based two-and three-dimensional models of the cell comprising separate compartments for cellular membrane and actin cortex, cytoskeleton, and nucleus. To the extent that such continuum models can capture stress and strain patterns within the cell, it can help relate biological influences of various types of force application and dynamics under different geometrical configurations of the cell.

Introduction

Cells can be modeled as continuum media if the smallest length scale of interest is significantly larger than the dimensions of the microstructure. For example when whole-cell deformations are considered, the length scale of interest is at least one or two orders of magnitude larger than the distance between the cell's microstructural elements (namely, the cytoskeletal filaments), and as such a continuum description may be appropriate.

ABSTRACT: This chapter attempts to develop a general perspective on the phenomenon of active protrusion by ameboid cells. Except in rare special cases, principles of mass and momentum conservation require that one consider at least two phases to explain the process of cellular protrusion. These phases are best identified with the cytosolic and cytoskeletal components of the cytoplasm. A continuum mechanical formalism of Reactive Interpenetrative Flows (RIF) is contructed. It is general enough to encompass within its framework a large range of theories of active cell deformation and movement. Most physically plausible theories of protrusion fall into one of two classes: protrusion driven by cytoskeletal self-interactions; or protrusion driven by cytoskeletal–membrane interactions. These are described within the RIF formalism. The RIF formalism is cast in a form that is amenable to computer simulations through standard numerical algorithms. An example is next given of the numerical study of the most elementary protrusive event possible: the formation of a single pseudopod by an isolated round cell. Some final thoughts are offered on the role of modeling in understanding cellular mechanical activity.

Cellular protrusion: the standard cartoon

Over the past decades, experiments have consistently demonstrated that active protrusion in animal cells is accompanied by a local increase in cytoskeletal density through active polymerization. A review of the evidence for this is beyond the scope of this chapter, but key points include the high concentration of filamentous actin observed by fluorescence or EM at the leading edge of growing protrusions, as well as the abrogation of protrusion by nearly any disruption of actin polymerization, such as that caused by cytochalasin.

Although the importance of the cytoskeleton in fundamental cellular processes such as migration, mechanotransduction, and shape stability have long been appreciated, no single theoretical or conceptual model has emerged to become universally accepted. Instead, a collection of structural models has been proposed, each backed by compelling experimental data and each with its own proponents. As a result, a consensus has not yet been reached on a single description, and the debate continues.

One reason for the diversity of opinion is that the cytoskeleton plays numerous roles and it has been examined from a variety of perspectives. Some biophysicists see the cytoskeleton as a cross-linked, branched polymer and have extended previous models for polymeric chains to describe the actin cytoskeleton. Structural engineers have drawn upon approaches that either treat the filamentous matrix as a continuum, above some critical length scale, or as a collection of struts or beams that resist deformation by the bending stiffness of each element. Others observe the similarity between the cell and large-scale structures whose mechanical integrity is derived from the balance between elements in tension and others in compression. And still others see the cytoskeleton as a gel, which utilizes the potential for phase transition to accomplish some of its dynamic processes. Underlying all of this complexity is the knowledge that the cell is alive and is constantly changing its properties, actively, as a consequence of many environmental factors. The ultimate truth, if indeed there is a single explanation for all the observed phenomena, likely lies somewhere among the existing theories.

As with the diversity of models, a variety of experimental approaches have been devised to probe the structural characteristics of a cell.

ABSTRACT: Most plant and animal cells possess a complex structure of filamentous proteins and associated proteins and enzymes for bundling, cross-linking, and active force generation. This cytoskeleton is largely responsible for cell elasticity and mechanical stability. It can also play a key role in cell locomotion. Over the last few years, the single-molecule micromechanics of many of the important constituents of the cytoskeleton have been studied in great detail by biophysical techniques such as high-resolution microscopy, scanning force microscopy, and optical tweezers. At the same time, numerous in vitro experiments aimed at understanding some of the unique mechanical and dynamic properties of solutions and networks of cytoskeletal filaments have been performed. In parallel with these experiments, theoretical models have emerged that have both served to explain many of the essential material properties of these networks, as well as to motivate quantitative experiments to determine, for example concentration dependence of shear moduli and the effects of cross-links. This chapter is devoted to theoretical models of the cytoskeleton based on polymer physics at both the level of single protein filaments and the level of solutions and networks of cross-linked or entangled filaments. We begin with a derivation of the static and dynamic properties of single cytoskeletal filaments. We then proceed to build up models of solutions and cross-linked gels of cytoskeletal filaments and we discuss the comparison of these models with a variety of experiments on in vitro model systems.

Introduction

Understanding the mechanical properties of cells and even whole tissues continues to pose significant challenges. Cells experience a variety of external stresses and forces, and they exert forces on their surroundings – for instance, in cell locomotion.