Keywords

Introduction

Biopolymers such as nucleic acids, proteins, and polysaccharides play diverse biological functions and are components of various materials. Nucleic acids encode hereditary information and instructions for protein synthesis. In addition, the unique hybridization properties of nucleic acids provide building blocks of nanomaterials, nanomachines, and nanosensors [1–13] and are considered as a viable platform for highly parallel biological computing [14–18]. Proteins mainly perform enzymatic reactions and participate in cellular signal transduction and communication but also play critical structural and mechanical roles (e.g., supporting cell shape and elasticity) and are natural components of bioadhesives, biocomposites [19, 20], and bio-fibers like collagen [21] and silk [22–24]. Natural, synthetic, and hybrid proteins have recently been exploited for development of new biomaterials with rationally tuned elastic properties [25–29]. Polysaccharides, either alone or as components of glycoproteins or peptidoglycans that are exploited for energy storage, participate in molecular recognition between biomolecules and also play important structural roles, e.g., in the cell wall of plants and bacteria and as components of hydrogels and biofilms [30, 31]. They are also components of many natural and semisynthetic materials (e.g., paper, cotton, rayon).

The mechanical properties of individual biomacromolecules and their nanostructures are critically important for their biological and other functions. For example, the mechanics of the DNA double helix plays an important role during cell division, DNA replication, DNA damage repair, and transcription of DNA information onto RNA [32]. The mechanics of protein networks, such as present in the extracellular matrix, is essential for cell shape, cell flexibility, and binding interactions between cells (cell adhesion) [33–70]. Also, muscle elasticity is partially determined and regulated by the elastic properties of giant modular proteins [71–76]. The mechanics of polysaccharide chains (such as cellulose chains and fibers) is important in providing rigidity to cellular structures (e.g., wood), and the flexibility of sugar rings is exploited in molecular recognition between sugars and lectins and is important for enzymatic reactions such as glycolysis [77]. The combined mechanics of proteins and polysaccharides is exploited in biohydrogels such as those lubricating joints [78–81]. Since these biomacromolecules are also used as building blocks for various nanostructures and nanomachines, the characterization of their mechanical properties is of considerable significance to nanotechnology. The progress in directly measuring mechanical properties of individual biomacromolecules paralleled the development of a variety of single-molecule visualization, manipulation, and characterization techniques. This chapter will briefly introduce the most popular single-molecule manipulation techniques and will review the nanomechanical properties of individual biomacromolecules determined using these methods.

Polymer Elasticity and Techniques to Study the Mechanical Properties of Single Polymer Molecules

The mechanical properties of individual biomacromolecules are typically examined by means of single-molecule force spectroscopy (SMFS) techniques [73, 82–89]. In SMFS, individual macromolecules or their fragments are attached to a substrate and to a force probe and stretched (by separating the two), and their extension and tension are accurately measured [90]. The relationship between the applied force (tension) and extension that describes molecule’s elasticity has been coined a force spectrogram. Biomacromolecules covered in this chapter are polymeric in nature so they are composed of many identical or similar units (monomers). For this reason, the primary source of their elasticity is entropic in origin [91–94]. The entropy is at its maximum in equilibrium and is gradually decreased when the polymer ends are separated and the monomers are forced to align with the direction of the stretching force. Fully stretched polymers would have just a single configuration so their configurational entropy would be zero and attaining such a state would require an infinite force [94, 95]. However, even before such high forces are generated entropically, the chemical and physical bonds within the polymeric structure gradually extend according to their own stiffness and the polymer exhibits the enthalpic elasticity that results in continually increasing the contour length of the polymer (overstretching) [73, 96]. The enthalpic elasticity may also manifest itself as an abrupt transition in the force–extension relationship typically in the form of a force peak or a force plateau, when individual bonds or their groups undergo a discrete conformational force-induced transition that results in an abrupt lengthening of the polymer [73, 83, 96–99].

Atomic Force Microscopy (AFM)

Advantages: Excellent length resolution, low-high force range (5–10,000 pN), and constant-velocity or constant-force conditions are available, no need for specialized attachment.

Disadvantages: Cantilever spring constants are difficult to determine accurately [100, 101], not suitable for probing events at low forces (<5 pN).

AFM was invented in 1986 by Binning, Quate, and Gerber [102, 103] on the basis of an earlier invention of the scanning tunneling microscope (STM) [104, 105]. AFM was initially applied primarily as an imaging tool, but soon its power for mechanical manipulation of individual biomolecules was realized [106]. In SMFS measurements by AFM, molecules are attached at their termini or at random positions to a substrate and to the AFM tip, either specifically through chemical bonds or using ligand–receptor specificity (e.g., avidin–biotin) or even nonspecifically through physisorption [90]. The molecules that formed a bridge between substrate and the tip may be stretched in solution, which is of significance to measurements on biomacromolecules. The stretching process is controlled by means of a highly precise piezoelectric actuator that moves the sample away from the AFM tip or vice versa. The force experienced by the molecule (its tension) is determined through monitoring the bending of the AFM cantilever, which is followed by a split photodiode that measures the position of a laser beam reflected off of the cantilever and projected onto the diode. Force and length resolutions of SMFS measurements by AFM are on the order of 1 pN and < 1 nm. The main advantages of AFM as a force spectrometer are its superb length resolution, the ability to stretch short molecules, and the ability to apply small forces (piconewton order) and uniquely also very large forces (tens of nanonewtons). Also, AFM force spectrometers are fast and allow large loading rates (force/time) that are of importance when studying lifetimes of intermolecular bonds. SMFS by AFM are typically carried out under constant extension rate [73] or force clamp conditions [107].

Optical Tweezers

Advantages: Excellent force resolution and excellent length resolution at low forces (0.1–100 pN).

Disadvantages: Requires functionalized biopolymers to tether to bead

Optical tweezers [108–111] use a focused laser light to create a potential well that traps dielectric objects, as first observed by A. Ashkin in 1970 [112]. An appropriately surface-functionalized micron-size dielectric bead (e.g., coated with avidin) can be used to attach to it a terminally functionalized biopolymer (e.g., biotin-labeled DNA) and can be captured by an optical trap. The other end of the molecule can be attached to a surface or to another bead kept, e.g., in a glass pipette by suction. The molecule is stretched by moving the surface or the second bead away from the optical trap (e.g., by means of piezoelectric actuator). A microscope-based video system accurately monitors the position of the first bead relative to the center of the optical trap to determine the applied force, while the translation of the second bead is accurately measured to determine the molecule extension. Optical tweezers provide superb force and length resolution on the order of <0.1 pN and <1 nm and are widely used in SMFS of DNA and proteins.

Magnetic Tweezers

Advantages: Excellent force resolution at low and medium forces (0.01–100 pN), simple method for applying torque

Disadvantages: Requires functionalized biopolymers to tether to bead

Magnetic tweezers use micron-size superparamagnetic beads, which develop a net magnetic moment in an external magnetic field and are pulled by a magnetic force that is proportional to the field gradient [113–115]. Similar to optical tweezers, a molecule of interest can be tethered between a surface (bead) and a superparamagnetic bead and stretched by an external magnetic field. Forces on the order of 0.01–100 pN can be easily exerted by magnetic tweezers [94]. In addition to stretching, magnetic tweezers provide a very simple means to apply a torque to the molecule of interest allowing it to be rotated and coiled [113, 116–118].

Biomembrane Force Probe

Advantages: High precision in spring constant measurement, excellent force range (0.01–1,000 pN)

Disadvantages: Limited to probing molecules that appear on (or are introduced to) the cell surface

In the biomembrane force probe assay (BFP), a small glassy bead is biochemically “glued” to a pressurized membrane capsule (e.g., red blood cell membrane) that is held by a pipette through a controlled amount of suction [41]. Different negative pressures result in different membrane tension, so the probe stiffness can be easily controlled (by the pressure) and forces on the order of 0.01–1,000 pN can be generated. The bead itself is decorated at low surface density with molecules of interest that are brought to contact to their cognate receptors, presented on another cell. By forming contacts between the bead and the surface of the investigated cell, specific bonds between ligands (presented on the bead) and receptor (presented on the cell surface) are formed and then ruptured by moving the bead away from the cell surface. In this way bonds’ strength and lifetimes can be accurately measured. Forces and extensions are determined via optical microscopy by the amount of deformation of the membrane transducer and the position of the glassy bead. Typical resolution is <0.5 pN and <5 nm [41, 119].

Nanopore Techniques

Advantages: Capabilities to uniquely measure size and charge of molecules

Disadvantages: Method not fully developed

Molecules pass through a nanometer-size pore in a membrane separating two compartments to which a potential gradient is applied and transiently block the ionic current flowing through the pore producing characteristic current blockage fingerprints. These current patterns can be used to infer various molecular properties of the traversing molecules, such as size and charge. It was proposed that natural or solid-state nanopores could be used to sequence long DNA strands, because characteristic blockage currents are different for different nucleobases [120–122]. Electric field-driven passage of charged biomacromolecules such as nucleic acids or uncharged but terminally functionalized with a charged “leader” (e.g., a short piece of DNA) molecules such as proteins can also be used to examine mechanical properties of traversing molecules. This is because in most cases these molecules are too bulky to pass through the pore and need to be stretched and unfolded before they will fit into a narrow pore [123, 124]. For direct measurements of the force applied to a molecule traversing a nanopore, its end can be attached a bead whose position can be accurately monitored in a force measuring optical trap [122].

Flow Techniques

Advantages: Readily simulate physiological flow conditions

Disadvantages: Requires careful calibration of flow to determine forces

Mechanical properties of biomacromolecules can also be studied by stretching them in an elongational flow [125, 126]. This can be achieved either directly due to the coupling of the flowing fluid with the molecule of interest or indirectly by attaching one end of the molecule to a surface and the other to a micron-size bead which then experiences a hydrodynamic force [127]. Also, to limit unwanted interactions between the molecule and the surface during flow measurements, a magnetic bead attached to the molecule of interest can be levitated magnetically while subjected to a horizontal hydrodynamic force [128].

Particle Tether

Advantage: Simple, inexpensive system suitable to various microscopy methods. The system does not involve external forces.

Disadvantage: Low spatial resolution

The tethered particle motion experiments (TPM) were first started by Jeff Gelles and colleagues in the early 1990s to study transcription by single RNA polymerase molecule [129, 130]. This method has been used to study DNA looping [131–142], DNA transposition [143], promoter sequences bending [144], and site-specific recombination [145, 146]. In a typical tethered particle experiment, a single polymer molecule is tethered between the microscope coverslip surface and a microsphere through specific binding [147]. Brownian motion of the bead is restricted to a semispherical region by the tethered polymer molecule and can be captured by an optical microscopy. Variance in travelling scope of the particle gives information about change in the length of the tethered polymer. TPM has the advantage of simple implementation, easy combination with optical and magnetic tweezers, and straightforward data analysis methods. However, TPM has low time resolution caused by the time cost by the probe to explore the region limited by the polymer tether [148]. Attempts to improve the accuracy of TPM include investigation of the volume effects of the bead [149, 150], suppression of the Brownian motion of the bead [151], simultaneous tracing of hundreds of single molecules by biochip [152], and development of proper data analysis approaches to obtain reliable kinetic parameters from TPM measurements [148, 153, 154].

A detailed comparison of these various single-molecule manipulation techniques along with the description of their advantages and limitations can be found in a number of review articles [155–159].

Applications

In its force measuring mode, AFM is typically used to stretch and relax DNA, proteins, and sugars either in isolation or also on living cells [83, 86, 87, 160] to study their elasticity, mechanical, unfolding, refolding, and binding behaviors. At low forces (<20 pN) AFM force spectra capture the characteristic highly nonlinear entropic elasticity of biomacromolecules. At higher forces various deviations from the purely entropic elasticity are frequently observed [82]. These deviations are indicative of structural and conformational transitions induced by force that on the experimental time scale are either reversible or irreversible. For example, using AFM-based SMFS the elasticity of individual titin molecules that govern the passive elasticity of muscle was characterized in various force regimes [73]. It was found that the entropic alignment of titin immunoglobulin and fibronectin-type domains occurs at low stretching forces, and at higher forces, these domains reversibly unfold providing an extra length to the muscle when needed [75]. In addition AFM is frequently used to probe the strength of the interactions between various biomolecules including receptors and ligands pairs [43, 60, 161]. Optical tweezers are frequently used to examine the elasticity of biopolymers at low forces, and OT measurements can be set up to exploit the nanomechanical properties of biomacromolecules (such as DNA) [96] to study the mechanochemical behaviors of various enzymes that process these molecules (e.g., DNA and RNA polymerases) [162]. OT measurements can also be used to follow near equilibrium folding/unfolding behavior of proteins, either alone [163] or while interacting with ligands [164]. Magnetic tweezers found many applications to study torsional elastic properties of DNA and to follow the work of special DNA enzymes that affect coiling properties of DNA (such as gyrases) [165, 166]. BFP techniques were found particularly suited for measuring receptor–ligand interactions on live cells [167]. Nanopore techniques are used to study folding properties of biomacromolecules and to examine the interactions between various biomolecules, and they are continuously improved for DNA sequencing applications [122, 123, 168]. Flow techniques are used for biopolymer elasticity measurements and in conjunction with fluorescence video microscopy are being applied to follow the interactions between various biomacromolecules (e.g., DNA–protein interactions) [128].

Atomic Force Microscopy (AFM)

AFM Instrumentation

The schematic of an AFM instrument is shown in Fig. 33.1.

Fig. 33.1
figure 1

(a) Schematic of an AFM instrument and (b) closeup of the cell containing the cantilever and probing the substrate. A laser probes the cantilever deflection which is detected using the difference between the top and bottom of a quadrant photodetector. The cantilever is suspended in a quartz cell over a substrate (clean glass or freshly evaporated gold) that has a drop of solution with the molecule of interest. A piezoelectric stack on the bottom controls the 3D movement of the substrate to contact the cantilever and then move away at constant force or constant velocity. Nonspecific attachment will allow the molecule (red lines) to attach to the cantilever and stretch the molecule of interest

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The principle of the AFM is conceptually simple: a small cantilever is first calibrated and then deflection of the cantilever during the stretching of an attached molecule is measured to precisely determine forces (from one to thousands of piconewtons) using Hooke’s law. Cantilever deflection is measured and recorded by tracking voltage signal output from multi-segment photodiode detector (quadrant detector module in most recent design). The final signal used to convert to force recording is

$$ \mathrm{V}=\Delta {\mathrm{V}}_{\mathrm{BT}}/\Sigma $$

where ΔVBT is the voltage difference between top and bottom area of the photodiode and Σ is the voltage sum from both areas. V is usually multiplied by an operational amplifier to improve the signal to noise ratio. The position of the sample is accurately controlled by a piezo actuator via a feedback–control loop with 0.2–0.5 nm resolution. These piezo actuator stages are usually equipped with capacitive or strain-gauge position sensors. The sensor signal output from the piezo controller is converted into distance using the voltage constant of the piezo,

$$ \Delta z=C\Delta V $$

where C is the constant measured and given by piezo actuator factory specifications, Δz is the movement of the piezo, and ΔV is the sensor voltage signal output of the piezo actuator.

Cantilever calibration is based on thermal noise method (one of dynamic deflection methods proposed by Hutter and Bechhoefer [169]). In this method, the cantilever and the tip are treated together as a simple harmonic oscillator with one degree of freedom. Thermal fluctuations are considered as the only motion of the oscillator with the Hamiltonian

$$ H=\frac{p^2}{2m}+\frac{1}{2}{k}_c{q}^2. $$

According to the equipartition theorem,

$$ <\frac{1}{2}{k}_c{q}^2>\kern-0.3em =\frac{1}{2}{k}_BT $$

where kB is the Boltzmann’s constant, kC is the spring constant of the oscillator, T is the absolute temperature, and q is the displacement of the oscillator. Therefore, kC can be obtained by measuring the mean-square spring displacement < q2 > due to thermal fluctuations at room temperature. This measurement is performed in the frequency domain by taking the power spectral density of the fluctuations of the photodiode signal δV (Fig. 33.2 is a representative power spectrum).

Fig. 33.2
figure 2

A typical power spectral density curve of the photodiode signal thermal fluctuations. Frequencies are shown in their natural logarithm (dB) and signals are displayed in Vrms. Integration interval is between the black lines

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The integration is performed according to Parseval’s equality

$$ <\delta {V}^2>\kern-0.3em =\frac{{\displaystyle \underset{a}{\overset{b}{\int }}\delta {V}^2(t) dt}}{{\displaystyle \underset{a}{\overset{b}{\int }} dt}}=\frac{{\displaystyle \underset{a^{,}}{\overset{b^{\hbox{'}}}{\int }}F{T}^2\left(\delta V\right) df}}{ ENBW} $$

where FT(δV) is the Fourier transform of δV and ENBW is the equivalent noise bandwidth of the spectrum. Integration of the power spectrum is usually done in an interval close to the resonance frequency of the cantilever such as depicted in Fig. 33.2.

To finally convert the photodiode voltage signal into force, a force spectrum is acquired by moving the sample vertically using the piezo, while the position of the piezo and resulting cantilever deflection are recorded simultaneously (Fig. 33.3).

Fig. 33.3
figure 3

Force spectrum. Voltage difference ΔV and piezo movement Δz used for slope calculation are shown in the graph

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Then the slope of the deflection versus piezo position is

$$ \mathrm{slope}=\Delta z/\Delta V $$

Thus, the voltage signal from the photodiode, V, is interpreted to force by the following formula

$$ F={k}_c\ast V\ast \mathrm{slope}=\frac{k_BT}{<\delta {V}^2>\ast {\mathrm{slope}}^2}\ast V\ast \mathrm{slope} $$

where T is room temperature (usually 300 K).

Sample Preparation

The sample is prepared simply by depositing the molecule of interest in the relevant substrate. Substrates commonly used for AFM single-molecule force spectroscopy experiments are gold or glass. In the most basic experiment, molecules attach to the surface and the tip nonspecifically. Since the attachment is nonspecific, the location that the molecule absorbs to the tip and substrate is random. To circumvent the random attachment, there have been methods developed to control attachment to the surface and tip, like thiol chemistry [170], HaloTag7 immobilization [171], Strep-Tag immobilization [172], and Ni–NTA functionalization [173]. In any kind of immobilization, it is important to have a positive control to differentiate between single- and multi-molecular AFM stretching experiments. Generally, only about 1 % of the data is usable. For protein unfolding experiments, a positive control can be designed by flanking the unknown protein of interest by previously characterized protein with known properties so that their presence indicates the recording is of a single molecule of interest (e.g., flanking unknown proteins by I27 domains of titin, which have a characteristic unfolding force of ∼200 pN and a contour length increment of ∼28 nm).

Stock solutions containing biomacromolecule (i.e., DNA, protein, polysaccharide) are usually diluted to 10–1,000 nM and incubated on the substrate for a period of time ranging from a few minutes to overnight. Appropriate incubation time and substrate choice are empirical and the ideal incubation case would allow the formation of a monolayer of the molecule on the substrate. Usually proteins are incubated on gold or functionalized glass for half an hour, DNAs are incubated on gold for more than 4 h, and polysaccharides are incubated on glass overnight. After incubation, the samples are usually washed several times before used for AFM pulling experiments to remove excess molecules that are not tethered to the surface of the substrates.

Experimental Procedure

AFM pulling experiments are carried out by gently moving the substrate relative to the cantilever tip through voltage applied to the piezo. The piezo can either control the height of the substrate relative to an immobilized cantilever or control the height of the cantilever relative to an immobilized substrate. Here we describe the procedure using nomenclature for the former method (as depicted in Fig. 33.1). There are two modes of motion for each pulling cycle (Fig. 33.4). In the up mode, initially the tip rests above the surface; the pulling measurement starts with the substrate moving up first to bring into contact with the tip under a voltage ramp generated by the computer and then descending to the original position. While in the down mode, at first the tip presses slightly onto the substrate; the substrate begins to move down to leave the tip; and after that, the substrate reverts to the origin. The stretching traces in both modes are obtained when the substrate departs from the tip; accordingly the relaxing traces are acquired in the other half of the cycle.

Fig. 33.4
figure 4

Up mode versus down mode. Red arrow indicates the moving direction of the substrate; blue arrow indicates the process

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Sometimes after the first cycle of pulling experiment, the cantilever tip still holds the molecule, which can be judged by discrepancy of the stretching trace tail from the horizontal line, since loss of the molecule would generate a horizontal baseline at the end of the stretching trace. Then refolding experiments can be realized by decreasing the pulling size to the desired length to stretch and relax the molecule for another several cycles.

Force–extension curves obtained from the AFM pulling experiments are selected first with several criteria and later analyzed with freely jointed chain (FJC) [73] or worm-like chain (WLC) [174] model for polymer elasticity. The FJC model considers the polymer chain segments (Kuhn segments) to be statistically independent. Assume that the elastic response of the polymer to the applied external force is purely entropic, then the extension < x > (instant end-to-end distance of the polymer projected on the direction of the force) as a function of the applied force is

$$ <x>\kern-0.3em ={L}_c\left( \coth \frac{F{l}_k}{k_BT}-\frac{k_BT}{F{l}_k}\right) $$

where Lc is the contour length of the molecule, lk is Kuhn segment length of the polymer, T is the temperature, and kB is the Boltzmann’s constant. In reality, molecules are often overstretched so that enthalpic contributions to the extension originated from bending of covalent bond angles and elongation of covalent bonds have to be taken into account. The revised version of the FJC model is

$$ <x>\kern-0.3em ={L}_c\left( \coth \frac{F{l}_k}{k_BT}-\frac{k_BT}{F{l}_k}\right)\ \left(1+\frac{F}{k_{\mathrm{segment}}{L}_c/N}\right) $$

where ksegment is the so-called segment elasticity that includes all the enthalpic effects and N is the number of segments contained in the polymer chain.

The WLC model treats the polymer as an irregularly curved filament. In this model, the force versus extension < x > relation is given by

$$ F\left(<x>\right)=\frac{k_BT}{l_p}\left[\frac{1}{4}{\left(1-\frac{<x>}{L_c}\right)}^2+\frac{<x>}{L_c}-\frac{1}{4}\right] $$

where Lc is the contour length of the molecule and lp is the persistent length of the polymer which equals one-half of lk. In AFM studies, proteins are usually described using the WLC model, while DNA and sugar are depicted by revised FJC model.

The selection criteria for the force extension curve use a combination of heuristics. Often a reference fingerprint pattern of the unfolding for a protein is used. The fingerprint is obtained from a recording that has a number of force–extension recordings with enough unfolding events of the flanking protein handles and the correct initial contour length before the first unfolding. The theoretical value for initial contour length which precedes the first unfolding peak of the reference protein can be calculated out by the estimated length of the proteins when in their native form.

Protein Mechanics

Protein Unfolding

The sequence of amino acids in a protein encodes the unique three-dimensional structure which is attained through folding. Properly folded proteins are important for their function. The determination of the folding pattern of proteins from the amino acid sequence to the 3D structure is an important problem in biology. Proteins are characterized by their 3D structure, their function, and also by their dynamic processes such as unfolding and folding rates and progression to the native state. The folding and unfolding processes are stochastic although not all conformational transitions are equally possible as the energy landscape (the space of all conformations and associated free energies) is not flat. Each protein state has an energetic contribution from configurational entropy and enthalpy from forming hydrogen bonding or electrostatic networks. The states of the protein are also subject to environmental factors such as temperature and concentration of denaturants and mechanical forces, which all contribute to the energy landscape. Since it is currently experimentally unfeasible to monitor all possible order parameters, the multidimensional landscape of protein folding is often studied by looking at a single-order parameter (e.g., N–C extension, GdmCl concentration, percent of native contacts) which then describes a small part of the entire energy landscape. When proteins are perturbed using force, such as using AFM-SMFS, the order parameter is along the extension of protein, between the tethered ends (usually N–C extension). The relevant parameters that characterize protein unfolding by AFM-SMFS are the unfolding force, the contour length increment, the unfolding rate, the refolding rate, and the distance to the transition state. These properties for a wide variety of proteins are tabulated in Table 33.1 and each of these properties is discussed below, in detail.

Table 33.1 Compiled parameters for the unfolding of various proteins. The Protein Data Bank (PDB) code indicates the representative structure used to determine native state lengths and native contacts and fold type. The mean unfolding force, Fu, is determined from the comparable loading rates. Proteins with information about intrinsic unfolding rates, ku 0; intrinsic folding rates (as determined by double-pulse protocol), kf 0; and the distance to transition state, xb, are included for respective proteins
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A protein, upon mechanical unfolding (Fig. 33.5) by AFM or other SMFS tool, adopts an unstructured chain of amino acids that behave in a worm-like chain manner in which their bonds tend to line up with the vector of the pulling direction with greater probability at higher force. Polyproteins will have several unfolding events as each protein domain contributes an unfolding event. The contour length distance between two unfolding events – as measured by the worm-like chain model – and the 3D crystal structure can be used to determine total extension of an amino acid at high force. For instance, the unfolding of a single I27 protein domain produces a contour length increment of 28.5 nm before the next unfolding event (Fig. 33.5). Before the rupturing of the single domain, it is assumed that the polyproteins have completely aligned with the pulling vector in their native state so that the extended chain distance is comprised of the increase in contour length (28.5 nm) plus the distance between the N-terminus and C-terminus of the protein in the native state, as measured by a NMR or X-ray crystal structure model (4 nm for IgI27). Thus, the total length 28.5 nm + 4 nm = 32.5 nm divided by the number of amino acids, 89, gives the distance of a fully extended single amino acid unit of IgI27, which is 3.65 Å. The mean length of a fully extended amino acid unit determined from the corresponding proteins in Table 33.1 with their corresponding PDB structures is 3.64 ± 0.04 Å (mean ± SE, n = 27). The consistency between these measurements then allows for the contour length increment, ΔLc, to be an important indicator of the number of amino acids unfolding within each unfolding event, if the protein structure is known, by inverting the previous calculation.

Fig. 33.5
figure 5

Typical force–extension plot of the unfolding of a polyprotein consisting of seven I27 domains from the titin protein (also called I91 domains). Each peak corresponds to an unfolding event of a single domain. The unfolding force for each domain is ∼200 pN. The dashed red line indicates a family of worm-like chain fits with a contour length spacing of 28.5 nm between unfolding events

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The unfolding force, Fu, for proteins ranges from as low as 5 pN up to 500 pN. The nonzero unfolding force is the result of proteins resisting unfolding due to an energy barrier between the unfolded and folded state along the particular pathway. The likelihood of unfolding increases exponentially with applied force because of thermal activation of bond rupturing so that the unfolding force is logarithmically dependent on the loading rate (cantilever stiffness x pulling velocity) [175]. The model derived from this concept, called the Bell–Evans–Ritchie model, interprets the log-linear dependence of the loading rate with force as an image of an energy barrier at a fixed location along the pathway. The intrinsic unfolding rate, ku 0, can then be determined by relating natural logarithm of the loading rate, r, to the most likely unfolding force, Fu, with the formula

$$ {F}_u(r)=\frac{k_bT}{x_b} ln\left(\frac{r{x}_b}{k_u^0{k}_bT}\right). $$

This model incorporates the parameter xb, which corresponds to the distance to the fixed location along the pathway from the unfolding state and the top of the barrier (the transition state). The experiments to determine these parameters are often referred to as “dynamic force spectroscopy” which simply involves performing pulling experiments at many loading rates (differing speeds and varying strengths of cantilever spring constants) to get enough data to reliably fit the parameters in the Bell–Evans–Ritchie model.

The intrinsic folding rate of proteins, kf 0, can be determined using a polyprotein and a “double-pulse” protocol. In this experiment, a polyprotein is unfolded during the first pulse and the number of unfolded modules determined. Then, after waiting a time t, a second pulse is applied and the number of modules unfolded is counted. The modules unfolded in the second pulse were able to refold during the time delay t. Thus, the proportion of the refolded protein modules out of the total unfolded modules in the first pulse can be plotted against the time delay t and fit to an exponential function to determine the intrinsic folding rate for each module, kf 0. The 3D structure and geometry of the pulling vectors also affects the unfolding force of proteins [176, 177]. However, most proteins unfolded from the N-terminus to the C-terminus have an unfolding force that correlates with contact density and their specific fold type (Fig. 33.6).

Fig. 33.6
figure 6

There is a correlation between the density of contacts and the unfolding force as determined by the Pearson correlation coefficient (PCC). Alpha-helical proteins exhibit the lowest unfolding forces and beta-folds exhibit higher unfolding forces but are geometry dependent

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It would be useful to determine the unfolding parameters of proteins from pulling experiments through computer simulations and purely theoretic means since many proteins already have a 3D structure available and experimental setups can be time intensive and costly. The atomistic detail of molecular dynamic simulations also provides insightful explanations for unfolding and folding phenomena. The analogous computer simulation to the experimental force spectroscopy experiment is steered molecular dynamics [255]. Steered molecular dynamics consists of a solvated protein model (determined from X-ray crystallography or NMR) that is fixed at one end and pulled at constant velocity or constant force on the other. These simulations do not reach the quasi-equilibrium conditions of the actual experiments because of limitations in computing time. Thus, the unfolding of proteins in steered molecular dynamics simulations are performed at speeds that are orders of magnitude faster. Table 33.2 lists the simulated peak unfolding force determined from the steered molecular dynamic simulations. Even though computer simulations are done at much higher speeds, there is a good correlation between the experimental results of the unfolding force and the theoretical unfolding force determined through simulation, as shown in Fig. 33.7.

Fig. 33.7
figure 7

Correlation between unfolding forces as performed by steered molecular dynamics (y-axis; from Table 33.2) and experimentally determined by atomic force microscopy (x-axis from Table 33.1). The correlation between the datasets is 0.84 determined by the Pearson correlation coefficient

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Table 33.2 Tabulated peak unfolding forces from steered molecular dynamic simulations with explicit water and comparable force fields (OPLS-AA or CHARM22 or AMBER99). Unfolding forces were compared at a pulling speed of 100 A/ns. Unfolding forces from simulations performed at other were extrapolated to 100 A/ns using unfolding force peaks from at least three different speeds fitted using log-linear regression (in accordance with Bell–Evans–Ritchie model where unfolding force depends on the logarithm of speed)
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Protein–Ligand Complex Unfolding

Proteins bind to cofactors and other proteins which can have an effect on their mechanical stability (as perturbed from the N to C extension) as tabulated in Table 33.3. In all cases the unfolding force is increased upon binding with the exception of titin kinase in which a new separate unfolding event occurred in the presence of ligand. Dr. Hongbin Li and colleagues exploited the difference between apo- and bound force spectra of proteins and measured the dissociation constant of proteins with their ligands by counting the proportion of bound proteins based on their unfolding force [30, 72, 73]. Surprisingly, the dissociation constants for proteins and their antibodies are lower than when measured with traditional experimental methods, while the dissociation constants for metal ions or small molecules are comparable. The differences between the SMFS measured dissociation constant and the bulk measured constant may be due to the mechanical perturbations required in SMFS, but more research in this area is needed.

Table 33.3 Protein unfolding characteristics when bound to their respective ligands. The fold increase indicates the increase in the mean unfolding force from the unfolding force tabulated in Table 33.1. The dissociation constant, Kd, was measured from SMFS experiments for some protein–ligand combinations and compared to the bulk measure dissociation constant (italics)
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Protein–Ligand Unbinding

The model for measuring the energy barrier between the folded state and the unfolded state of proteins can also be applied to the energy barrier between the bound and unbound state of proteins with their ligand. In these experiments the protein is conjugated to the tip and the ligand is conjugated to a surface. Pulling experiments are then performed and all the unbinding forces are recorded and tabulated in a histogram. The control experiment where there is no ligand conjugated to the surface serves as a distribution of nonspecific binding forces. Performing measurements at varying loading rates can determine the unbinding rate using the Bell–Evans–Ritchie model discussed in Section IIIA which can be used to extrapolate a dissociation constant for a protein and its ligand. The unbinding forces for several protein–ligand complexes and their dissociation constants (when known) are shown in Table 33.4. The unbinding forces range from 30 pN to 250 pN. The association constants (inverse of the dissociation constant) are positively correlated with the binding force as shown in Fig. 33.8.

Fig. 33.8
figure 8

The unbinding force at similar load rates for proteins and their ligands correlates with the bulk measured association constant (inverse of dissociation constant)

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Table 33.4 Tabulated unbinding forces for protein–ligand rupture events from SMFS. SMFS can also be used to determine the experimental dissociation constant, Kd, from single molecules perturbed by force (in contrast to the dissociation constant measured from a bulk sample in italic)
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Protein-Based Nanomaterials

The properties of proteins lend themselves to be useful building blocks for nanomaterials. Individual proteins can be selected based on mechanical strength and elastic characteristics and then fused at the DNA level into polyproteins. These polyproteins can then be linked via thiol chemistry or protein chemistry into biomaterials. Hongbin Li and colleagues tested this idea by building a protein-based elastomeric hydrogel ring constructed from a network of polyprotein GB1 domains [283]. The properties of the material can then be easily tuned by changing the composition of the polyproteins. Such materials are useful for developing scaffolds for tissue engineering.

DNA

The behavior of DNA under force has been studied for over 30 years using a variety of techniques. Single-molecule methods have allowed for precise characterization of DNA under force which revealed mechanical transitions that occur during unwinding and melting of the DNA helix [32, 284–292]. The origin of these transitions is still under study. One of the first DNA molecules studied is the λ phage dsDNA molecule, composed of 48,502 bp. When force is exerted on both ends of the molecule, it stretches and the force increases following the worm-like chain model very closely at forces below 50 pN. The force–extension profile is salt dependent, and measurements in 10 mM Na+ typically indicate a persistence length of ∼60 nm and an elastic modulus of ∼800 pN as shown in Table 33.5.

Table 33.5 Basic properties of λ phage DNA as studied by a variety of nanomolecular techniques
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The λ phage DNA, along with other types of double-stranded DNA and single-stranded DNA/RNA, has been shown to undergo overstretching transitions when stretched beyond 15 pN up to 1800 pN. The origin of these transitions is still a subject of study. The transition forces and gain in extension (as determined by normalized extensions) are shown in Table 33.6.

Table 33.6 Properties of nucleic acid polymers when perturbed by forces up to 1.8nN. Nomenclature: poly(dA) stands for a polymer of polydeoxyadenylic acid, or a single-stranded DNA molecule composed only of adenines whereas poly(A) stands for a polymer of polyadenylic acid, which consists of a single-stranded RNA molecule composed of adenines. Similarly, poly(dGdC)poly(dCdG) stands for a double-stranded DNA composed of CG repeats. Overstretching percent refers to the percent fraction of the plateau relative to the initial length
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Sugars

Entropic Elasticity and Force-Induced Conformational Transitions of Polysaccharides

AFM proved very valuable for characterizing the mechanical properties of many polysaccharides and enabled observations of unique force-driven transitions in the sugar rings [73, 83, 98, 308–310]. For AFM measurements, a polysaccharide sample is dissolved in water or another appropriate solvent at a wide range of concentrations ranging from 0.001 % to 20 % (w/w). A small drop of the solution (e.g., 50–100 μl) is placed on a clean substrate (glass, gold) and the molecules are allowed to adsorb to the substrate for several hours. The nonattached or weakly attached molecules are then removed from the surface by vigorous washing of the substrate. The molecules that strongly adsorbed to the substrate can then be lifted from it by the AFM tip and stretched in solution so their extension and tension can be accurately measured [90].

While some polysaccharides display the entropic elasticity that is typical of many structurally simple polymers at all forces (e.g., cellulose [308, 311]), some polysaccharides follow this behavior only at low forces, and at higher forces they show marked deviations from the entropic elasticity (e.g., amylose, dextran, pectin [83, 312]). These deviations are caused by force-induced conformational transitions within the sugar rings (e.g., chair–boat transitions in α-d-glucopyranose [98]), within the bonds connecting neighboring rings (e.g., bond flips in pustulan [313, 314]) or by force-induced separation of polysaccharide chains in multichain molecular structures (e.g., xanthan [315, 316]). Generally, when sugar monomers are connected by equatorial glycosidic linkages that lay in the plane of the sugar ring (as in cellulose), the elasticity of these polysaccharides is primarily entropic in nature and force spectra are quite simple. When monomers are connected by axial bonds (that are perpendicular to the plane of the sugar ring), the elasticity of those polysaccharides frequently displays interesting features (deviations) from the entropic elasticity that manifest themselves as pronounced force plateaus. Those plateaus were interpreted as indicative of force-induced transitions of the sugar ring from a low-energy conformation to a high-energy conformation that provides an increased separation of the consecutive glycosidic oxygen atoms and thus increases the contour length of the chain. Force bond rotations (flips) that occur over an energy barrier (such as in 1,6 linked polysaccharides) typically produce linear relationships between force and extension. Unwinding of helical structures (such as in xanthan) typically produces a long plateau in the force extension data and is typically associated with pronounced hysteresis between the stretching and relaxing part of the force–extension spectrum that reports strand separation in multiple helices. Table 33.7 compiles most of the known elasticity profiles (force spectrums) of various natural polysaccharides measured by AFM, in solution, on isolated molecules. The elasticity profiles of a number of polysaccharides measured directly on cell surfaces, from which they protrude, may be found through the references in a recent review [83].

Table 33.7 Table of sugars and their corresponding chemical structures and force spectra as determined by atomic force microscopy
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Outlook and Conclusion

The development of single-molecule manipulation techniques over the last 20 years enabled direct measurements of the mechanical properties of individual biomacromolecules. The number of biopolymers tested is steadily increasing and many fundamental observations and measurements were already repeated on the same systems by independent groups and verified. The data about types of elasticity and force-induced structural transitions obtained this way is invaluable for deciphering molecular mechanisms supporting life processes and for using these biopolymers in nanobiotechnology. Standardization of measurement conditions, automation of measurements [328] and analysis, and improvements of the accuracy of force sensors calibration [329] will continue to increase the quantity and reliability of single-molecule characterization measurements.