Single-Molecule Techniques

Derivation: The Trapping Force

For a dielectric sphere in the Rayleigh regime (particle radius $a \ll \lambda$), the trapping force has two components:

Gradient force (trapping):

The electric field of the focused laser induces a dipole moment$\mathbf{p} = \alpha \mathbf{E}$ in the bead, where$\alpha$ is the polarisability. The interaction energy is:

$$U = -\frac{1}{2}\langle \mathbf{p} \cdot \mathbf{E}\rangle = -\frac{\alpha}{2}\langle E^2 \rangle$$

The gradient force draws the bead toward the intensity maximum:

$$\mathbf{F}_{\text{grad}} = -\nabla U = \frac{\alpha}{2}\nabla \langle E^2 \rangle = \frac{2\pi n_m a^3}{c}\left(\frac{m^2 - 1}{m^2 + 2}\right)\nabla I$$

where $m = n_p/n_m$ is the ratio of particle to medium refractive indices, and $I$ is the laser intensity.

Scattering force (pushing):

$$F_{\text{scat}} = \frac{n_m}{c}\sigma_{\text{scat}} I, \quad \sigma_{\text{scat}} = \frac{128\pi^5 a^6}{3\lambda^4}\left(\frac{m^2-1}{m^2+2}\right)^2$$

Harmonic approximation near the focus:

Near the beam focus, the intensity is approximately Gaussian, so the gradient force becomes linear in displacement:

$$\boxed{F = -\kappa x}$$

where $\kappa$ is the trap stiffness (typically 0.01–1 pN/nm), determined by the laser power, bead size, and focusing optics. This makes the optical trap a calibrated force transducer.

Detailed Derivation: Polarisability of a Dielectric Sphere

The gradient force depends on the polarisability $\alpha$ of the particle. For a homogeneous dielectric sphere of radius $R$ with refractive index$n_p$ in a medium of refractive index $n_m$, the polarisability is derived from solving Laplace's equation with the boundary conditions at the sphere surface. The result is the Clausius-Mossotti relation:

$$\boxed{\alpha = 4\pi n_m^2 R^3 \left(\frac{m^2 - 1}{m^2 + 2}\right)}$$

where $m = n_p / n_m$ is the relative refractive index. For polystyrene beads ($n_p = 1.59$) in water ($n_m = 1.33$),$m = 1.20$ and the Clausius-Mossotti factor$(m^2 - 1)/(m^2 + 2) = 0.13$.

Physical origin: The electric field of the focused laser induces a dipole in the sphere. Since the field is non-uniform (focused), the forces on the positive and negative sides of the dipole do not cancel. The net force pulls the bead toward the region of highest field intensity (the focus):

$$\mathbf{F}_{\text{grad}} = \frac{n_m \alpha}{2}\nabla|\mathbf{E}|^2$$

Note that for trapping to work, we need $m > 1$ (the particle must be optically denser than the medium), so $\alpha > 0$, and the gradient force points toward the intensity maximum. For $m < 1$, the particle would be repelled from the focus.

Derivation: Worm-Like Chain (WLC) Model for DNA Force-Extension

Optical tweezers are widely used to stretch single DNA molecules. The force-extension behaviour is described by the worm-like chain (WLC) model, which treats DNA as a continuously flexible polymer with bending stiffness.

The WLC Hamiltonian for a polymer of contour length $L$ with tangent vector$\hat{t}(s)$ at arc length $s$ is:

$$\mathcal{H} = \frac{L_p k_B T}{2}\int_0^L \left(\frac{\partial \hat{t}}{\partial s}\right)^2 ds$$

where $L_p$ is the persistence length — the characteristic length scale over which tangent-tangent correlations decay:$\langle \hat{t}(s) \cdot \hat{t}(0) \rangle = e^{-s/L_p}$.

The exact force-extension relation for the WLC must be computed numerically. However, Marko and Siggia (1995) derived a remarkably accurate interpolation formula by matching the exact low-force (entropic) and high-force (enthalpic) limits:

$$\boxed{F = \frac{k_B T}{L_p}\left[\frac{1}{4\left(1 - \frac{x}{L}\right)^2} - \frac{1}{4} + \frac{x}{L}\right]}$$

Low-force limit ($x \ll L$): Expanding to first order in $x/L$:

$$F \approx \frac{k_B T}{L_p}\left[\frac{1}{4}(1 + 2x/L) - \frac{1}{4} + \frac{x}{L}\right] = \frac{3k_B T}{2 L_p}\cdot\frac{x}{L}$$

This is an entropic spring: force is proportional to extension ($F \propto x$), arising not from stretching chemical bonds but from the reduction of conformational entropy when the chain is extended. The effective spring constant is $k_{\text{entropic}} = 3k_B T / (2 L_p L)$.

High-force limit ($x \to L$): As the extension approaches the contour length, the first term dominates:

$$F \approx \frac{k_B T}{4 L_p}\cdot\frac{1}{\left(1 - x/L\right)^2}$$

The force diverges as $x \to L$. Physically, fully straightening a thermal polymer requires infinite force because it demands eliminating all thermal undulations, which would violate the third law of thermodynamics.

Experimental confirmation: Fitting to single-molecule stretching data for $\lambda$-phage DNA (48,502 bp, $L = 16.5$ $\mu$m) gives $L_p \approx 50$ nm for double-stranded DNA in physiological salt conditions. Single-stranded DNA has $L_p \approx 1$ nm, reflecting its much greater flexibility. At forces above $\sim 65$ pN, DNA undergoes an overstretching transition to a form ~1.7 times longer than B-DNA, not captured by the WLC model.

Cantilever Mechanics

The cantilever acts as a Hookean spring with stiffness $k_c$ (typically 10–100 pN/nm for biological applications). The deflection $d$ gives the force directly:

$$F = k_c \cdot d$$

Deflection is measured via the optical lever: a laser reflects off the cantilever onto a position-sensitive photodetector. The cantilever stiffness is calibrated using the thermal noise method (equipartition):

$$k_c = \frac{k_B T}{\langle d^2 \rangle}$$

Detailed Derivation: Force from Cantilever Deflection

The AFM cantilever is a microfabricated silicon or silicon nitride beam, typically 100–200 $\mu$m long, 20–40 $\mu$m wide, and 0.5–1 $\mu$m thick. Its spring constant follows from Euler-Bernoulli beam theory for a rectangular cantilever clamped at one end:

$$k_c = \frac{E w t^3}{4 L^3}$$

where $E$ is Young's modulus of the material, $w$ is the width,$t$ is the thickness, and $L$ is the length. The force on any object in contact with the tip is simply:

$$\boxed{F = k_c \times d}$$

where $d$ is the cantilever deflection, measured by reflecting a laser off the cantilever onto a four-quadrant photodiode (the optical lever method). This achieves sub-angstrom deflection sensitivity.

Detailed Derivation: Bell-Evans Model

Consider a molecular bond modelled as a particle in a potential well with barrier height $\Delta G^\ddagger$ and distance $x_\beta$ to the transition state. Without force, the spontaneous unbinding rate is (from Kramers theory):

$$k_0 = \nu \exp\left(-\frac{\Delta G^\ddagger}{k_B T}\right)$$

An applied force $F$ tilts the energy landscape, lowering the barrier by$F \cdot x_\beta$. The force-dependent unbinding rate becomes:

$$\boxed{k(F) = k_0 \exp\left(\frac{F x_\beta}{k_B T}\right)}$$

This is the Bell model (1978). Under a constant loading rate $r = dF/dt$, the force increases linearly with time:$F = r \cdot t$. The survival probability of the bond obeys:

$$\frac{dS}{dt} = -k(F(t)) \cdot S(t) = -k_0 \exp\left(\frac{r \cdot t \cdot x_\beta}{k_B T}\right) S(t)$$

The rupture force probability density is $p(F) = k(F) S(F) / r$. Setting$dp/dF = 0$ to find the most probable rupture force:

$$\boxed{F^* = \frac{k_B T}{x_\beta}\ln\left(\frac{r \cdot x_\beta}{k_0 \, k_B T}\right)}$$

This is the Evans-Ritchie result (1997). Key predictions:

• • $F^*$ increases logarithmically with loading rate $r$
• • The slope of $F^*$ vs. $\ln(r)$ gives $k_B T / x_\beta$, yielding the barrier width
• • The intercept gives the intrinsic off-rate $k_0$
• • A change of slope at high loading rates indicates a second, inner barrier

Typical values for biomolecular bonds: $x_\beta \sim 0.1\text{--}1$ nm,$k_0 \sim 10^{-3}\text{--}10$ s$^{-1}$,$\Delta G^\ddagger \sim 10\text{--}30\, k_B T$.

Derivation: FRET Efficiency

The rate of energy transfer from donor to acceptor at distance $r$ is:

$$k_T(r) = \frac{1}{\tau_D}\left(\frac{R_0}{r}\right)^6$$

where $\tau_D$ is the donor lifetime in the absence of acceptor and$R_0$ is the Förster radius. The $r^{-6}$ dependence arises from the dipole-dipole coupling.

The FRET efficiency is the fraction of donor excitation events that result in energy transfer:

$$\boxed{E = \frac{k_T}{k_T + k_D} = \frac{1}{1 + (r/R_0)^6}}$$

where $k_D = 1/\tau_D$ is the total donor decay rate. Note that$E = 0.5$ when $r = R_0$, making $R_0$ the "50% efficiency distance."

Experimental Observables

The FRET efficiency can be measured via:

• Intensity ratio:$E = \frac{I_A}{I_A + I_D}$ (corrected for cross-talk and direct excitation)
• Donor lifetime:$E = 1 - \tau_{DA}/\tau_D$ where $\tau_{DA}$ is the donor lifetime in the presence of acceptor
• Donor photobleaching rate:$E = 1 - k_{b,D}/k_{b,DA}$

Force and Torque

The force on a superparamagnetic bead in a magnetic field gradient is:

$$\mathbf{F} = \nabla(\mathbf{m} \cdot \mathbf{B}) \approx \frac{V \chi}{2\mu_0}\nabla B^2$$

where $V$ is the bead volume and $\chi$ is the magnetic susceptibility. The force is controlled by moving the magnets (changing the gradient).

Force calibration uses the fluctuations of the tethered bead. From the equipartition theorem applied to the transverse fluctuations $\delta x$ of a bead tethered by a molecule of length $l$:

$$F = \frac{k_B T \cdot l}{\langle \delta x^2 \rangle}$$

Rotating the magnets applies a torque to the bead, which twists the DNA molecule. The extension of DNA as a function of linking number (turns) reveals supercoiling behaviour: plectoneme formation causes a characteristic linear decrease in extension.

Detailed Derivation: Force on a Superparamagnetic Bead

A superparamagnetic bead contains nanoscale ferromagnetic grains embedded in a polymer matrix. In an external field $\mathbf{B}$, the grains align, giving the bead a net magnetic moment. The energy of a magnetic dipole $\mathbf{m}$ in a field is:

$$U = -\mathbf{m} \cdot \mathbf{B}$$

The force is the gradient of this energy:

$$\mathbf{F} = \nabla(\mathbf{m} \cdot \mathbf{B}) = (\mathbf{m} \cdot \nabla)\mathbf{B}$$

For a superparamagnetic bead, the magnetisation is proportional to the field (below saturation):

$$\mathbf{m} = \frac{\chi V}{\mu_0}\mathbf{B}$$

where $\chi$ is the volume magnetic susceptibility, $V$ is the bead volume, and $\mu_0$ is the permeability of free space. Substituting:

$$\boxed{F = \frac{\chi V}{2\mu_0}\nabla(B^2)}$$

The factor of $1/2$ arises because $\mathbf{m}$ itself depends on $\mathbf{B}$. In practice, the force is controlled by adjusting the distance between the magnets and the sample: closer magnets produce steeper gradients and larger forces (range: 0.01–100 pN).

Photon Statistics

A single fluorophore emits photons as a Poisson process with rate$I = \sigma \Phi I_{\text{exc}} / (h\nu)$, where $\sigma$ is the absorption cross-section, $\Phi$ the quantum yield, and$I_{\text{exc}}$ the excitation intensity. The number of photons detected in time $T$ follows:

$$n \sim \text{Poisson}(\eta \cdot I \cdot T)$$

where $\eta$ is the overall detection efficiency (typically 1–10%).

Photobleaching: Each fluorophore has a finite number of excitation-emission cycles before irreversible photodestruction. The photobleaching time follows an exponential distribution:

$$p(t_{\text{bleach}}) = k_{\text{bleach}} e^{-k_{\text{bleach}} t}$$

For single-step photobleaching, the intensity trace shows a sudden drop from a constant level to background — a hallmark of single-molecule detection (multiple molecules would show stepwise decreases).

Derivation: Kramers Escape Rate

Consider a Brownian particle in a one-dimensional potential $U(x)$ with a well at $x = 0$ and a barrier at $x = x_b$. The dynamics obey the overdamped Langevin equation:

$$\gamma \dot{x} = -U'(x) + \xi(t), \quad \langle \xi(t)\xi(t') \rangle = 2\gamma k_B T \,\delta(t - t')$$

Equivalently, the probability density $\rho(x,t)$ evolves according to the Smoluchowski (Fokker-Planck) equation:

$$\frac{\partial \rho}{\partial t} = \frac{1}{\gamma}\frac{\partial}{\partial x}\left[U'(x)\rho + k_B T \frac{\partial \rho}{\partial x}\right]$$

Kramers solved for the steady-state flux over the barrier using an absorbing boundary condition beyond $x_b$. Expanding $U(x)$ to second order around the well minimum ($U''(0) = \omega_0^2$) and the barrier top ($U''(x_b) = -\omega_b^2$), the escape rate is:

$$\boxed{k = \frac{\omega_0 \omega_b}{2\pi\gamma}\exp\left(-\frac{\Delta G^\ddagger}{k_B T}\right)}$$

where $\Delta G^\ddagger = U(x_b) - U(0)$ is the barrier height,$\omega_0 = \sqrt{U''(0)}$ characterises the curvature of the well, and$\omega_b = \sqrt{|U''(x_b)|}$ characterises the curvature of the barrier. The prefactor $\omega_0 \omega_b / (2\pi\gamma)$ is the attempt frequency, representing how often the particle "tries" to escape.

Protein Folding and Unfolding Trajectories

Optical tweezers and AFM have revealed that proteins unfold and refold through distinct intermediates invisible to ensemble measurements. By attaching a protein between two beads (or a bead and a surface) and applying a controlled force ramp, individual unfolding events appear as sudden extension increases ("rips") in the force-extension curve. Each rip corresponds to the unravelling of a structural domain.

Titin, the giant muscle protein, shows a characteristic sawtooth pattern with ~28 nm extension steps corresponding to individual immunoglobulin domains unfolding. The unfolding force ($\sim 200$ pN) and refolding force ($\sim 5$ pN) differ dramatically, revealing the hysteresis inherent in non-equilibrium pulling.

DNA-Protein Interactions

Single-molecule techniques have illuminated how proteins interact with DNA. Magnetic tweezers reveal real-time activity of topoisomerases as discrete changes in DNA linking number. Optical tweezers show RNA polymerase translocation along DNA at single-base-pair resolution, including pausing, backtracking, and arrest events. RecBCD helicase unwinds DNA at speeds up to 1,500 bp/s, with force-dependent unwinding rates that probe the enzyme's mechanochemistry.

Chromosome Mechanics

Magnetic tweezers and micropipette aspiration experiments on isolated chromosomes reveal remarkable mechanical properties. Mitotic chromosomes behave as elastic rods with a Young's modulus of $\sim 1$ kPa and can be stretched to several times their native length. The force-extension response reflects the hierarchical organisation of chromatin: nucleosome unwrapping at $\sim 5$ pN, 30 nm fibre unfolding at$\sim 20$ pN, and higher-order loop disruption at $\sim 50$ pN.