Model systems for optical trapping: the physical basis and biological applications

Abstract

The micromechanical methods, among which optical trapping and atomic force microscopy have a special place, are widespread currently in biology to study molecular interactions between different biological objects. Optical trapping is reported to be quite applicable to study the mechanical properties of surface structures onto bacterial (pili and flagella) and eukaryotic (filopodia) cells. The review briefly summarizes the physical basis of optical trapping, as well as the principles of calculating the van der Waals, electrostatic, and donor-acceptor forces when two microparticles or a microparticle and a flat surface are used. Three main types of model systems (abiotic, biotic, and mixed) used in trapping experiments are described, and the peculiarities of manipulation with living (bacteria, fungal spores, etc.) and non-spherical objects (e.g., rod-shaped bacteria) are summarized.

The theory of optical trapping at macro-, meso-, and microscopic levels of detail

Light acts significantly on micro- and nanometer objects comparable in size with the radiance wavelength. Light pressure was first hypothesized by German astronomer Johannes Kepler in 1619 to explain the observation that comet tails point away from the Sun. The term “light pressure” was promoted by James Maxwell based on the developed electromagnetic theory. After Heinrich Hertz’s experiments, electromagnetic waves were shown to be absorbed or reflected by solid bodies, so that radiance was expected to exert the mechanical pressure on a solid matter. Pyotr Lebedev experimentally discovered this phenomenon in 1899, but the measured value of pressure was very small.

In 1970, Arthur Ashkin proposed that high-intensity monochromatic radiance can be used for atoms trapping (Ashkin 1970; Essiambre 2021), and in 1986, he trapped and replaced silica microparticles in fluid by a tightly focused argon laser beam (λ = 514.5 nm) (Ashkin et al. 1986). The scheme of an optical trap is shown in Fig. 1.

Fig. 1.

Forces exerting on a colloid microparticle near a focused beam (left). The vector of light pressure acting on a particle upwards corresponds to the direction of the Poynting vector $\overrightarrow{S}$ (right). Here $\overrightarrow{B}$ and $\overrightarrow{E}$ are the vectors of magnetic and electrical fields; λ is the wavelength

Generally, the trapping (optical) force is mediated by light pressure (it is an equivalent to the scattering force (F_s) and proportional to the Poynting vector of optical field) and by the gradient force (F_grad) which is proportional to the gradient of light intensity (Feynman et al. 1963; Jing 2018). The gradient force directing toward the laser focus tends to attract the colloid particle to the region of highest light intensity, and its magnitude must exceed both light pressure and Brownian motion. This is usually achieved by using a high-aperture objective.

Magnitude of these forces can be calculated differently, and it depends on the ratio between the microparticle size (let it be the sphere with the radius r) and wavelength (λ). We will present theory comporting to three general size regimes namely:

• (i)Microscopic – r << λ
• (ii)Macroscopic – r >> λ
• (iii)Mesoscopic – r ~ λ

For the case of r << λ, the condition for Rayleigh scattering is met, and the microparticle can be considered a point dipole—so that the scattering force is proportional to light intensity as follows (Neuman and Block 2004):

$$\mathit{Fs}=\frac{I_0\sigma n_m}{c}$$ 1

$$\sigma =\frac{128{\pi }^5{r}^6}{3{\lambda }^4}{\left(\frac{m^2-1}{m^2+1}\right)}^2$$ 2

where I₀ is light intensity (the mean value of magnitude of the Poynting vector), σ is the scattering cross section for photons, c is the speed of light in vacuum, n_m is the refractive index for the medium, and m is the ratio of the refractive index for a particle (n_p) to the refractive index for the medium (m = n_p / n_m).

Magnitude of the gradient force averaged over the time is mediated by interaction of a dipole with the electromagnetic field and depends on the refractive index for a particle and its polarizability (α):

$$F_{\mathit{grad}}=\frac{2\mathit{\pi \alpha }}{{\mathit{cn}}_m^2}\nabla I_0$$ 3

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

where

As mentioned before, magnitude of the gradient force is proportional to the gradient of light intensity (gradI₀), so that it is greater than 0 at m > 1 (directed toward the focus) and less than 0 at m < 1 (directed away from the focus). If the radius of a microparticle is much greater than the wavelength (r >> λ), the laws of geometric optics can be applied. In such cases, incident radiation is a sum of discrete beams, each of which has definite intensity, direction, and polarization. Magnitude of two forces can be calculated using the angles of incidence (θ) and diffraction (φ) (Ashkin 1992) as follows:

$$F_s=\frac{{\mathit{Pn}}_m}{c}\left\{1+Rcos2\theta -\frac{T^2\left[cos\left(2\theta -2\phi \right)+Rcos2\theta \right]}{1+R^2+2Rcos2\phi }\right\}$$ 5

$$F_{\mathit{grad}}=\frac{{\mathit{Pn}}_m}{c}\left\{Rsin2\theta -\frac{T^2\left[sin\left(2\theta -2\phi \right)+Rsin2\theta \right]}{1+R^2+2Rcos2\phi }\right\}$$ 6

where P is the radiant power, n_m is the refractive index for the medium, c is the speed of light in vacuum, and R and T are dimensionless coefficients of scattering and transparence showing a part of energy that is dissipated or transmitted after falling the beam on the surface. Parameters T and R depend on the refractive index and the angles of incidence and diffraction.

It is clear that direction of the force and its magnitude depend on the refractive index, the matter of particles, the number of beams forming the focus, equitability of beams intensities, and the angle of incidence. It was established experimentally (Ashkin 1992) that for particles with n = 1.2, an increase of θ from 30° to 60° causes a corresponding four-fold rise in the scattering force and 2.5-fold rise in the gradient force.

This is in reasonable agreement with measurements for large (Chang et al. 2006) and small (Ashkin 1992; Liu et al. 2015) spheroids. In spite of the fact that objects with r > 10λ are not often used in optical trapping studies, the theoretical model combining the principles of geometric optics and Newtonian mechanics may be used as an explanation for the angular momentum of a microparticle (Fig. 2).

Fig. 2.

The origin of the gradient force acting on a spherical particle near a non-focused (left) and focused (right) laser beam. Photons having finite momentum and energy (black arrows, beams a, b) pass through a microparticle with the definite refraction index which is different from this parameter for a medium. The scattering force (red arrows) is directed upwards, in the same direction as photons. After transmitted beams diffracted (beams a′, b′), their momenta are changing (dp_a and dp_b, respectively) and the gradient force directing laterally (toward the energy maximum, left) or axially (right). Light intensity profiles are shown as Gaussian

Mathematically, the most complicated is the model describing the behavior of a mesoscopic particle (r ~ λ) at the focus of the beam, since in this case neither the laws of geometric optics nor the concept of a point dipole is applicable. This kind of modeling is important for biological studies in which size of objects (commercial microsphere preparations, pro- and eukaryotic cells, vesicles etc.) is in the range between 0.1 and 10 λ. There are some theories based on the principles of classical electrodynamics and tensor calculations (Zemánek et al. 2002; Almaas and Brevik 1995), but a unifying concept is yet to be developed.

According to the generalized Lorentz-Mie theory, the total optical force is derived from the conservation law of electromagnetic wave momentum and can be computed by evaluating the flux of the Maxwell stress tensor through any virtual surface S enclosing the object:

$$F_i={\oint }_S{T}_{\mathit{ij}}\cdot n_j\mathit{dS}$$ 7

where n_j is the unit vector normal to this enclosing surface S and T_ij is the Maxwell stress tensor.

Given that, the stress tensor of the magnetic field is expressed as:

$${T_{\mathit{ij}}}^{\mathit{MF}}=\frac{1}{{\mu }_0}\left(B_i{B}_j-\frac{1}{2}{\delta }_{\mathit{ij}}{\left|\mathrm{B}\right|}^2\right)$$ 8

and the stress tensor of the electric field as:

$${T_{\mathit{ij}}}^{\mathit{EF}}={\epsilon }_0\left(E_i{E}_j-\frac{1}{2}{\delta }_{\mathit{ij}}{\left|\mathrm{E}\right|}^2\right)$$ 9

The Maxwell stress tensor becomes (Neves and Cesar 2019):

$$T_{\mathit{ij}}={\epsilon }_0{\epsilon }_b{E}_i{E}_j\ast +{\mu }_0{H}_i{H}_j\ast -\frac{1}{2}\left({\epsilon }_0{\epsilon }_{b}E\cdot E\ast +{\mu }_{0}H\cdot H\ast \right){\delta }_{\mathit{ij}}$$ 10

The formula (7) can be rewritten as (Neves and Cesar 2019):

$$F_i=\frac{1}{2}Re{\oint }_S\left[{\epsilon }_0{\epsilon }_b{E}_i{E}_j\ast +{\mu }_0{H}_i{H}_j\ast -\frac{1}{2}\left({\epsilon }_0{\epsilon }_{b}E\cdot E\ast +{\mu }_{0}H\cdot H\ast \right){\delta }_{\mathit{ij}}\right]n_j\mathit{dS}$$ 11

where Re is the Reynolds number, ε₀ is the absolute dielectric permittivity of vacuum, ε_b is the relative permittivity of the background dielectric medium in which the incident field propagates, μ₀is the magnetic permittivity of vacuum, and δ_ijis the Kronecker delta.

Given that the total fields outside the sphere can written in terms of the incident (index “inc”) and scattered (index “inc”) fields E = E_inc + E_s, H = H_inc + H_s, the total optical force can be recovered as below (Salandrino et al. 2012):

$$\begin{array}{l}{F}_i=-\frac{\sqrt{{\epsilon }_b}}{c}\frac{1}{2}Re\underset{r\to \infty }{lim}\underset{S}{\oint }\left[\left(E_s+E_{\mathit{inc}}\right)\times \left(H_s^{\ast }+H_{\mathit{inc}}^{\ast }\right)-E_{\mathit{inc}}\times H_{\mathit{inc}}^{\ast }\right]\mathit{dS}\\ =-\frac{\sqrt{{\epsilon }_b}}{c}\frac{1}{2}Re\left[\underset{r\to \infty }{lim}\underset{S}{\oint }\left(S-S_{\mathit{inc}}\right)\mathit{dS}\right]\end{array}$$ 12

where c is the speed of light in vacuum and S and S_inc are the total Poynting vector and the incident field Poynting vector, respectively.

Description of forces acting on and between particles in an optical trapping experiment

In the previous section, we described the forces acting on a particle within the optical trap due to radiation pressure exerted from the focused light. However as the optical trap drags the particle through solution, the particle will experience resistance due to the frictional force exerted by the viscous solvent. Here we describe friction at the colloid level of theory that is suitable for the optical trapping of spherical and non-spheroidal microparticles. Non-spheroids often encountered in biological systems (e.g., rod-shaped bacteria, mitochondria, chloroplasts) also can be trapped, but their hydrodynamic characteristics are different from those for spheroids. For the sphere moving at constant velocity in the medium, the force of viscosity can be calculated from the equation:

$$F=v\cdot \gamma$$ 13

where v is the particle’s velocity (m/s) and γ is the drag coefficient—γ = 6πηvr. Here η is the medium viscosity ((N·s)/m²), and r is the radius of a particle.

Note: linear relationship between the drag force acting on the microsphere and its velocity may be applied for calibration of the optical trap through stiffness coefficient. In modern optical tweezers, the calibration is carried out through the power spectrum density estimation (Svoboda and Block 1994a, b).

Therefore, when a uniform motion of a spherical object occurs, the force of viscosity is equal in every direction. Due to the ratio between the length and width for a cylindrical particle is more than 1, a force diagram in this case is more complex and described by two drag coefficients—transverse (γ_⊥) and longitudinal (γ_∥) (Chang et al. 2006):

$${\gamma }_{\perp }\sim \frac{4\mathit{\pi \eta L}}{ln\frac{L}{r}}$$ 14

$${\gamma }_{\Vert }\sim \frac{2\mathit{\pi \eta L}}{ln\frac{L}{r}}$$ 15

where η is the liquid viscosity, r is the radius of a cylinder, and L is the length of a cylinder.

The refractive indexes for biological objects are similar to ones for polystyrene (1.56) and for silica (1.45). Thus, its value is about 1.38 (1.37 for a cytoplasm and 1.39 for a nucleus) for eukaryotic cells (line CHO) (Chang et al. 2006). Mean refractive indexes for 2.2 nm flagellin layer from Salmonella typhimurium flagella is 1.43 (Bahadoran et al. 2014) and for Escherichia coli cells is 1.39 (Valkenburg and Woldringh 1984). However, it should be noted that the optical force is maximal when abiotic particles are used. For example, trap stiffness is 1.7–2.9 times higher for polystyrene bead in comparison with CHO cells and 4 times higher than for rod-shaped forms of Klebsiella pneumoniae (Chang et al. 2006). This is very much in line with Liang’s results (Liang et al. 2000) in which the optical force was 3-fold increased when polystyrene microspheres (d = 1 μm) were used (E. coli cells were as reference objects). Nevertheless, despite some objects (e.g., spheroids and rod-shaped bacteria) can differ in their shape or size, the values of trap stiffness are very similar allowing to use bacterial cells as objects for trapping with known limitations (see below). It is important to note that rod-shaped bacteria grown in vitro vary in form from cylinders to spheroids.

Forces acting on the objects irregularly moving in the medium (e.g., movable bacteria, ciliary microalgae, and protozoa) can be calculated based on the conception of motion organelles as harmonic oscillators (Brenner and Winet 1977).

Physicochemical basis of short-distance interactions between colloidal surfaces in a liquid medium

A liquid phase, for which saline buffers with physiological osmolarity are mostly used, is essential for optical trapping when a biological experiment is in process. In case of mineral substrates using, the modeling of short-distance interactions between a carrier (microsphere or living object) and a surface, which target molecules are linked to, is of particular importance.

In terms of physical chemistry, contacting surfaces and a surrounding fluid can be considered typical condensed phases. The term “condensed phase” refers to a state of any system, in which the number of structural elements (atoms, molecules, ions etc.) is extremely large, and they are packed relatively close together. In such a case, there are four forces which are significant for molecular interaction: electrodynamic (van der Waals forces), electrostatic (Coulomb forces), acid–base interactions, and the Brownian motion component. Total energy of molecular interactions in colloid systems includes free energies of four mentioned interactions (Israelashvili 2011a).

Xu Zhou et al. (2017) considered methods of calculating the energies in dependence on geometries of interacting surfaces in detail. The most common theory describing interactions between colloid particles (bacteria or spheroids) and a solid substrate is DLVO (on the surnames of scientists—Derjaguin, Landau, Verwey, Overbeek). Van Oss (1994) made a broader description of this theory. In accordance with it, full energy of interactions between two objects (G^total) can be represented as a sum:

$$G^{\mathit{\text{total}}}=G^{\mathit{LW}}+G^{\mathit{EL}}+G^{\mathit{AB}}+G^{\mathit{Br}}$$ 16

where G^LW is the free energy of van der Waals interactions (Lifshitz–van der Waals interactions), G^EL is the free energy of electrostatic interactions, G^AB is the free energy of Lewis acid–base interactions (acid−base component), and G^Br is the free energy of Brownian motion.

In spite of widespread application of optical trapping in molecular biology and biophysical studies, only a few model systems are in use—these are spheroids or cylinders in most cases. Depending on the form of a microparticle and its surface characteristics, each component of total energy should be calculated in different ways.

Lifshitz–van der Waals forces

The Lifshitz−van der Waals interfacial free energy component depends on the physicochemical properties and geometry of interacting surfaces. The most typical and commonly used variants are shown in Fig. 3.

Fig. 3.

Calculation of the van der Waals force (F^LW) depending on the geometries of interacting surfaces—for a sphere (A), a cylinder (B), and for two spheres (C). Notes: r is the radius of a sphere, D is the separation distance between two planar surfaces (m), and A is the Hamaker constant (its value for polystyrene in water is 1.3×10^–20 J) (Israelashvili 2011a)

Describing the adhesiveness of the material implies the force of interaction, but the energy parameter is commonly used. The Lifshitz−van der Waals interfacial free energy component of interaction between the spherical object (bacterium or sphere) (index 1) and the flat surface (index 2) separating by a thin water layer (index w) in accordance with van Oss (1994) can be calculated by the equation:

$$\mathrm{\Delta }{G}_{1w2}^{\mathit{LW}}=\frac{A}{6}\left[\frac{2r^2}{D\left(4r+D\right)}+\frac{2r^2}{{\left(2r+D\right)}^2}+ln\frac{D\left(4r+D\right)}{{\left(2r+D\right)}^2}\right]$$ 17

where A is the Hamaker constant, r is the radius of a sphere, and D is the separation distance between two planar surfaces.

The Lifshitz–van der Waals interaction force between the bacteria and membrane surface in the aqueous media is calculated upon differentiation (F = – dG/dD):

$$F_{1w2}^{\mathit{LW}}=2\pi \mathrm{\Delta }{G}_{1w2}^{\mathit{LW}}\frac{D_0^{2}r}{D^2}$$ 18

The Hamaker constant is defined as a function of distance between microobjects:

$$A=12\pi D_0^2\mathrm{\Delta }{G}_{1w2}^{\mathit{LW}}$$ 19

where D₀ is the equilibrium cut off distance (its experimentally determined value is 0.157 nm (van Oss 1994)) and $\mathrm{\Delta }{G}_{D0}^{\mathit{LW}}$ is the Lifshitz−van der Waals interfacial free energy component for two plates (indexes 1 and 2), separated by a thin water layer (index w) with thickness D₀.

Finally, $\mathrm{\Delta }{G}_{1w2}^{\mathit{LW}}$ is calculated based on values of coefficients describing total surface energy of particles:

$$\mathrm{\Delta }{G}_{1w2}^{\mathit{LW}}=-2\left(\sqrt{{\gamma }_w^{\mathit{LW}}}-\sqrt{{\gamma }_2^{\mathit{LW}}}\right)\left(\sqrt{{\gamma }_w^{\mathit{LW}}}-\sqrt{{\gamma }_1^{\mathit{LW}}}\right)$$ 20

where ${\gamma }_1^{\mathit{LW}}$, ${\gamma }_2^{\mathit{LW}}$, and ${\gamma }_w^{\mathit{LW}}$ are dispersive components of total surface energy for the substrate (index 1), a particle (index 2), and water (index w).

Electrostatic interactions are important for distant adhesion of bacterial cells at the distance more than Coulomb barrier. If the barrier has been overcome due to either the influence of hydrodynamic forces or cells motility, the Lifshitz−van der Waals forces are becoming more significant. The last ones are mediated by thermodynamic properties of interacting surfaces and the surrounding medium. It is known that the distance, on which the van der Waals component is emerging, does not exceed hundreds of angstroms, resulting the stable interaction between an antigen and an antibody is proposed to be conditioned primarily by this one (Chen and Zhu 2004).

Electrostatic interactions

Colloid particles in the liquid medium (in water or saline buffers) have a surface charge. Most solid particles (including microbial cells) are negatively charged at neutral pH values. This property causes local redistribution of solvent ions, resulting in opposite charged ions located near the particle surface, and charges of the same polarity are distributed at some distance apart the colloid. This leads to the formation of the double electric layer around a particle, and approaching of these colloids leads to their mutual repulsion.

Free energy of electrostatic interaction between a spheroid (bacterium or microsphere) (index 1) and a flat surface (index 2) separated by a thin water layer (index w) as it was described earlier (van Oss 1994) depends on geometry of interacting surfaces. In case when potentials of particles are constant, this dependence can be expressed by the formula:

$$G_{1w2}^{\mathit{EL}}={\mathit{\pi \epsilon }}_r{\epsilon }_{0}r\left[2{\psi }_{01}{\psi }_{02}ln\left(\frac{1+e^{-\mathit{\kappa D}}}{1-e^{-\mathit{\kappa D}}}\right)+\left({\psi }_{01}^2+{\psi }_{02}^2\right)ln\left(1-{\mathrm{e}}^{-2\mathit{\kappa D}}\right)\right]$$ 21

From this equation, the electrostatic force can be obtained:

$$F_{1w2}^{\mathit{EL}}={\mathit{\pi \epsilon }}_r{\epsilon }_{0}r\left({\psi }_1^2+{\psi }_2^2\right)\frac{2\kappa }{exp\left(2\mathit{kD}\right)-1}$$ 22

where ε₀ is the permittivity of vacuum (8.854×10^–12 C/(V·m)), ε_r is the relative dielectric permittivity of water (78.55 at 25 °С), κ is the inverse Debye screening length (960×10^–9 m for pure water, 0.3×10^–9 m for 1М water solution of sodium chloride (Bhushan 2008)), ψ₀₁ and ψ₀₂ are the surface potentials of the particle surface, r is the radius of a particle, and D is the separation distance between two surfaces.

The potentials ψ₀₁ and ψ₀₂ are calculated based on the empirically determined zeta potentials for particles or matters:

$${\psi }_0=\zeta \left(1+\frac{D}{r}\right)e^{\mathit{\kappa D}}$$ 23

where ζ is the zeta potential.

The Debye screening length is determined using the equation:

$$\frac{1}{k}=\sqrt{\frac{{\mathit{\epsilon \epsilon }}_0{k}_{B}T}{2n_i{v}^2{e}^2}}$$ 24

where e is the electron charge (1.6 · 10^–19 C), n_i is the number of ions per m³, v is the valence of ions, k_B is the Boltzmann constant, and T is the absolute temperature.

Formula 18 is used when the surface potential of the particle does not exceed 25 mV (Oliveira 1997), but it was noted (Rajagopalan and Kim 1981) that it is possible to use it up to the potential 60 mV.

Acid−base interaction

The more matter polar are, the more significant van der Waals forces are for total surface potential (γ). For this reason, for non-polar liquids (e.g., alkanes) and some solid matters (Teflon, polyethylene), total potential is equal to the Lifshitz–van der Waals potential, so that γ = γ^LW. In turn, for non-polar matters (both liquid and solid), the acid–base component (γ) is significant, too. In such case, the equation for the total surface potential would be written as γ = γ^LW + γ^АВ.

Generally, the acid–base component can be determined via the equation ${\gamma }^{\mathit{АВ}}=2\sqrt{{\gamma }^{+}{\gamma }^{-}}$; in so doing, electron donating (γ⁻) and accepting (γ⁺) components of total surface energy would depend on geometry of interacting surfaces and their chemical composition (see Table 1).

Table 1.

Surface energy coefficients for some matters, liquids, and particles

Object or matter	Surface energy parameters (mJ/m2)			References
	γLW	γ+	γ–
B. cereus 569 (spores)	29.8 ± 0.4	0	0	Xu Zhou et al. (2017)
B. subtilis PS832 (spores)	33.0 ± 1.4	0	21.0 ± 1.4
B. subtilis (cells)	34.0	0	79.8	Thwala et al. (2013)
Escherichia coli XA90 (cells)	35.3	0.16	67.33	Liu and Zhao (2005)
Pseudomonas fluorescence (cells)	36.2	N/d	N/d	Chen and Zhu (2004)
Glass	43.7 ± 0.7	0.27	61	Xu Zhou et al. (2017)
Mica	35.8	2.9	111.9	Thwala et al. (2013)
Polystyrene	42	0	1.1	Morgan and Wilkie (2007)
Water	21.8	25.5	25.5	Docoslis and Giese (2000)
Glucose (solution)	42.2	34.5	85.6

Notes: γ^LW is a dispersive, γ⁻ is an electron donating, and γ⁺ is an accepting component of total surface energy. Standard deviations are given for experimentally established parameters. If the parameter was determined theoretically, there is only its calculated value. N/d no data

The acid–base interfacial free energy component of interaction between two surfaces separated by a thin water layer can be calculated as shown:

$$G{}_{1w2}{}^{\mathit{AB}}=2\mathit{\pi r\lambda }\mathrm{\Delta }{G}_{1w2}^{\mathit{AB}}exp\left(\frac{D_0-D}{\lambda }\right)$$ 25

where λ is the decay length. This parameter for pure water is about 0.2 nm, and its value in saline buffers is comparable with the diameter of hydrated ions (up to 1.8 nm) (van Oss 1994).

As in the case of the van der Waals interaction, the acid−base component ($\mathrm{\Delta }{G}_{1w2}^{\mathit{AB}}$) follows from:

$$\mathrm{\Delta }{G}_{1w2}^{\mathit{AB}}=2\left[\sqrt{{\gamma }_w^{+}}\left(\sqrt{{\gamma }_1^{-}}+\sqrt{{\gamma }_2^{-}}+\sqrt{{\gamma }_w^{-}}\right)+\sqrt{{\gamma }_w^{-}}\left(\sqrt{{\gamma }_1^{+}}+\sqrt{{\gamma }_2^{+}}-\sqrt{{\gamma }_w^{+}}\right)-\sqrt{{\gamma }_1^{+}{\gamma }_2^{-}}-\sqrt{{\gamma }_1^{-}{\gamma }_2^{+}}\right]$$ 26

Finally, the acid–base force of interaction becomes:

$$F{}_{1w2}{}^{\mathit{AB}}=2\mathit{\pi r}\mathrm{\Delta }{G}_{1w2}^{\mathit{AB}}exp\left(\frac{D_0-D}{{\lambda }_{\mathit{AB}}}\right)$$ 27

Brownian motion component

Calculated free energy of Brownian motion (G^Br) for particles comparable in size with a microbial cell at room temperature is about 0.4 × 10^–20 J (Liu and Zhao 2005) and may be considered negligible in total energy of interaction.

Steric interactions

Steric interactions are generally applied to polymer-coated surfaces, and for this reason, they are significant to biological systems, especially when a microorganism is used as an object. Bacterial adhesion to the surfaces coated with biopolymers (lipopolysaccharides, polysaccharides, teichoic and lipoteichoic acids, glycoproteins), some of which are considered polyelectrolytes, can also be assessed.

The potential energy of steric interactions includes several components. Currently, some complementary mathematical models were proposed for their explanations (Oliveira 1997), but none of them has been confirmed empirically. The force of steric interactions (F^steric) for two polymer-coated surfaces can be calculated by the formula (de Gennes 1987):

$$F^{\mathit{\text{steric}}}=k_B{\mathit{TS}}_{\mathit{ads}}^{3/2}\mathit{Dr}\left[{\left(\frac{2L_0}{D}\right)}^{9/4}-{\left(\frac{D}{2L_0}\right)}^{3/4}\right]$$ 28

where k_B is the Boltzmann constant and T is the absolute temperature; S_ads is the adsorption density; r is the radius of a particle; D is the separation distance between two surfaces, the bacterium (or microsphere) and the substrate; and L₀ is the equilibrium thickness of the polymer brush.

Hydrophobicity

Hydrophobicity of one or another matter implies the value of the wetting angle. Nevertheless, the commonly agreed scale has not been investigated yet, because the wetting angle depends on physicochemical properties of the surface (chemical composition and relief). For a smooth surface, the angle θ is calculated as given by the Cassie equation (Israelashvili 2011b).

$$cos\theta =\sqrt{{\mathrm{f}}_1^2{cos}^2{\theta }_1+{\mathrm{f}}_2^2{cos}^2{\theta }_2}$$ 29

where f₁ and f₂ = 1 – f₁ are the parts of hydrophobic and hydrophilic groups and θ₁ and θ₂ are the wetting angles for hydrophilic and hydrophobic components.

For the matter with hydrophobic groups only and θ = 120°, cos θ₁ = 0.5, for ideally hydrophilic surfaces (θ = 0°) cos θ₁ = 1. If the matter is amphiphilic, which means that it has parts that are both hydrophilic and hydrophobic in the ratio 1:1 (f₁ = f₂ = 0.5), the wetting angle is calculated as cos θ = 0.5 × 1 – 0.5 × 0.5 = 0.25, yielding θ ≈ 76°. It is obvious that the angle for hydrophobic matters will be between 90° < θ < 180°, and for hydrophilic ones, it will be less than 90°.

Despite an importance of hydrophobicity for many types of biological interactions—intra- and intermolecular (DNA – proteins, protein – protein) and intercellular (models “pathogen – host cells”)—a uniform concept for the mechanisms of these interactions has not yet been established (Israelashvili 2011b). Experimental studies with use of surfaces coated with surfactants (Sun 2017) confirm the fact that hydrophobic forces prevail over the van der Waals interactions at the distance of 10–15 nm apart the surface, reaching maximum values at short distances (about 1 nm).

Influence of some factors on the interaction force

The most difficult in experimental modeling is the particle shape parameter. To simplify the calculations, in most cases, the shape of a microobject is taken as a sphere, even in cases when it differs from that (e.g., rod-shaped bacteria, yeast cells, bacterial spores). Here the next approximating equation can be applied (Xu Zhou et al. 2017):

$$R_{\mathit{\text{spore}}}={\left(\frac{4D^3+6D^2+\left(L-D\right)}{32}\right)}^{1/3}$$ 30

where D is the diameter of a large hemisphere (forming the ends of a spore) and L is the largest length of a spore.

To avoid undesirable binding of bacterial cells with the substrate, it is possible to coat the surface with molecules similar in physicochemical properties to those on the surface of the microparticles. Bacterial cells adhere more strongly to hydrophobic surfaces than to hydrophilic ones (Fletcher et al. 1983); therefore, additional adsorption of fatty acids or hydrocarbons onto the hydrophilic substrate (glass, metal) enhances cells attachment. In addition to lipopolysaccharides (Wang et al. 2015) and teichoic acids (Swoboda et al. 2010), some surface structures—flagella (Haiko and Westerlund-Wikström 2013), fimbriae (Kline et al. 2009; Berne et al. 2015), and curli (Squeglia et al. 2018)—can play a role in this process. The adhesive properties of a complex biopolymer are mediated primarily by physicochemical characteristics of its components. Lipopolysaccharides (LPSs) of Gram-negative bacteria can be taken as the examples of such substances. The full form of LPS (lipid A, core, O-side polysaccharide chains) is characterized as neutral (Farley et al. 1988; Walker et al. 2004; Boyer et al. 2011) or negatively charged (Knirel et al. 2006; King et al. 2009) weak polyelectrolyte. Rough LPS containing lipid A and core only and lacking O-side chains is more hydrophobic than the full form of LPS (Boyer et al. 2011).

Bivalent cations (e.g., Ca²⁺, Mg²⁺) can also mediate adhesion between two negatively charged surfaces as linkers. Thus, it was confirmed that adhesion of Streptococcus mutans and Fusobacterium nucleatum cells to titanium surfaces is much stronger when calcium ions were added (Badihi Hauslich et al. 2013). The unified theory that explains this fact has not been developed yet. Nevertheless, cationic bridges (Marshall et al. 1971) or local precipitation of microbial polymers (Rutter 1980) are also possible, but influence of other cations has not been well studied. It was shown in case of Pseudomonas fluorescence adhesion to copper, brass, and aluminum that interaction force decreasing may be mediated by toxic effect of Zn²⁺ and Cu²⁺ ions (Vieira et al. 1993).

Influence of pH and ionic strength of solution is mainly due to a changing of surface potentials of studied objects. The increase of H⁺ concentration leads to zeta potential decreasing and consequently to increasing of electrostatic forces in total adhesiveness. Ions H⁺ may also have a great influence on activity of adsorbed enzymes and structure of many proteins; in some cases, pH shifting causes conformational changes in macromolecular structure, dramatically decreasing their sorption to the substrate. Thus, adhesion of Mycobacterium smegmatis cells to aluminum oxide decreased after the medium was acidified (Smith et al. 2019).

Optical tweezers-based model systems used in biological studies

Applications of optical trapping for measurement of molecular interactions have expanded considerably at present. For that purpose, different model systems are used. They can be conditionally divided into three groups: abiotic (Fig. 4A, B), biotic (Fig. 4C, D), and mixed (Fig. 4E–J). The first two types are commonly used. Abiotic models are based on mineral (silica, modified mica, metals) or polymeric compounds, on which target molecules (polysaccharides, proteins, etc.) are linked by chemical modification. In modern optical tweezers, the function of precision displacement (horizontally or vertically) is implemented, so that the force and energy of molecular interaction can be measured. One of the primary benefits of using abiotic surfaces is a possibility of reliable chemical modification by target molecules with certain concentration. On the other hand, there is a high rigidity of interacted surfaces not requiring use of complex elastic binding models. In addition, standardized carriers (microbeads having a certain diameter) are used in many cases, making the calibration procedure much more convenient.

Fig. 4.

Optical trapping models used in biological experiments. Abiotic: A – “bead (microsphere) – surface,” B – “bead – bead.” Biotic: C – “pathogen – host cell,” D – “bacterium – bacterium.” Mixed: E – “bacterium – surface,” F – “bead – bacterium on a surface,” G – “bead – bacterium on a bead,” H – “bead – gliding bacterium,” I – “bead – encapsulated bacterium,” J – “bead – eukaryotic cell.” Notes: Ag antigen, Ab antibody

Abiotic models

Depending on the experimental conditions and technical potential of optical tweezers, two types of models can be constructed:

• (i)“Bead – bead”
• (ii)“Bead – substrate”

In the first case, molecules (antigen or antibodies) are adsorbed onto the inert or chemically modified (-NH₂, -COOH, and other groups) bead surface, and then the objects are approached and interacted for a limited period of time after which they are retracted at the start position. One of the beads is optically trapped in a laser beam focus, a micropipette or a second trap fixes the other one (usually of a larger diameter), or it can be adhered to a substrate. Two bead models are most often used in experiments to study protein folding or molecular interactions between proteins, nucleic acids, and carbohydrates (Choudhary et al. 2019). If the second component of a model system is the surface with linked molecules, the experimental scheme is not fundamentally different from described previously but with the difference that the microobject is approached and retracted vertically (Konyshev et al. 2020). Given that interacting surfaces are rigid and abiotic models are more stable in time, they can be used for the estimation of intermolecular forces in molecular pairs (e.g., “antigen – antibody”).

Streptavidin–biotin complexes are the most well-characterized. This is primarily due to the highest affinity of the molecules, whose spatial and thermodynamic properties are well-studied. It should be noted that registered intermolecular forces depend on loading rate, number of molecular pairs, and geometry of interacting surfaces and may vary in a wide range even if standard methods are used. Thus, the rupture force between streptavidin and biotin assessed by AFM is 150–160 pN when biotin molecules are adsorbed onto agarose microspheres (Moy et al. 1994; Florin et al. 1994) and 340 pN if the mica surface is used as a substrate (Lee et al. 1994). When loading rate is varying, the interaction force is in the wider range, from 5 to 170 pN, and the mean force is 160 pN (Merkel et al. 1999). The values of the interaction force for several molecular pairs, estimated using optical tweezers or atomic force microscopy, are given in Fig. 5.

Fig. 5.

The semi-logarithmic diagram of forces registered with use of optical tweezers (black dots) and by atomic force microscopy (white dots). All dots are numbered in the diagram from “1” to “44” with step of 5. The dots’ numbers correspond to the numbers of articles as listed below: 1, Formosa-Dague et al. 2018; 2, Zhou et al. 2012; 3, Svoboda and Block 1994a, b; 4, Reck-Peterson et al. 2006; 5, He et al. 2019; 6, Junker and Rief 2009; 7, Liang et al. 2000; 8, Stout 2001; 9, Burgos-Bravo et al. 2018; 10, Simpson et al. 2003; 11, Wang et al. 1998; 12, Jass et al. 2004; 13, Ziegler et al. 2016; 14, Keller et al. 2018; 15, Crick et al. 2014; 16, Stangner et al. 2013; 17, Allen et al. 1997; 18, Björnham et al. 2009; 19, Becke et al. 2018; 20, Li et al. 2018; 21, Håti et al. 2015; 22, Smith et al. 2001; 23, Mashaghi et al. 2014; 24, Stout 2001; 25, Lim et al. 2017; 26, Mártonfalvi et al. 2017; 27, Willemsen et al. 1998; 28, Maier et al. 2003; 29, Florin et al. 1994; 30, Castelain et al. 2016; 31, Chuartzman et al. 2017; 32, Vinckier et al. 1998; 33, Hinterdorfer et al. 1996; 34, Valotteau et al. 2017; 35, Peñaherrera 2015; 36, Sedlak et al. 2019; 37, Vinckier et al. 1998; 38, Peñaherrera 2015; 39, Schoeler et al. 2015; 40, Peñaherrera 2015; 41, Valotteau et al. 2017; 42, Yip et al. 1998; 43, Vitry et al. 2017; 44, Milles et al. 2018.

Interpretation of experimental results is related to identifying the specific interactions in a pair “antigen – antibody.” However, even if highly purified preparations and abiotic systems that are relatively stable in time are used, non-specific interactions may also be detected.

Biotic models

Living objects (e.g., pro- and eukaryotic cells) as the components of the model systems are rarely used primarily because of their instability in time and heterogeneity of their chemical composition. Nevertheless, they are essential for direct visualization of phagocytosis when the pathogen (native or fluorescent) is positioned near the target cell membrane (Tam et al. 2010; Tam et al. 2011; Bain et al. 2015). The last fact emphasized the contribution of membrane gangliosides and their complexes with other lipid molecules in total adhesiveness of Histoplasma capsulatum cells—it has been shown that lipid rafts can stabilize adhesion and mediate further phagocytosis of the pathogen by macrophages (Guimarães et al. 2019).

The imaging techniques not only allowed visualization of phagocytosis but detailed the changes in both participants (intracellular vesicles transport, mitosis inhibition, etc.). Thus, previously unknown membrane structures in dendritic cells (so named “fungipodia”) were detected by optical trapping—presumably they play a significant role at early stages of fungal pathogens neutralization (Bain et al. 2015; Neumann and Jacobson 2010). Similar experiments were done with use of AFM (El-Kirat-Chatel and Dufrêne 2012).

Dynamic experiments allow measuring the interaction force between a pathogen and a target cell directly, but due to the instability of eukaryotic cells, these approaches are not widespread. There are only the limited data about estimation of the rupture force in this way (Crick et al. 2014; Fedosov et al. 2011). Combined investigations are much more promising when optical trapping is used together with the techniques of molecular inhibition and changing the molecular structure of cytoskeleton (Crick et al. 2014) or Raman spectroscopy (Sinjab et al. 2020).

Biotic models can be used for characterization of molecular interactions at early stages of biofilms formation (Fig. 4D), playing an important role in bacterial colonization of living tissues and artificial materials. In this case, two traps are used, each of which contains a bacterium. Precise approaching of the trapped oscillating cells and comparison of their oscillating amplitudes can establish the significance of a particular component in primary contact.

Models in which direct interaction between the eukaryotic cell and viral particles is measured may also be categorized as biotic. The main challenge in implementing this model is the fact that sub-micron particles are used here, so that they cannot be directly visualized. In the simplest case, the virion can be considered a nucleoprotein which can be adsorbed onto a mineral surface (e.g., microsphere) (Sieben et al. 2012). Models in which the viral particle is adhered to the microsphere through a tether-like linker (two-stranded DNA molecule bound with streptavidin) are also known. Streptavidin interacts with biotin, and the latter is linked to the transferrin receptor that is integrated into the viral supercapside. Another end of the DNA molecule is linked to a polystyrene microsphere covered by antidigoxigenin antibodies (Schimert and Cheng 2018). This three compound system (“microsphere – tether – virion”) allows the indirect manipulation of the viral particles near the surface of a eukaryotic cell. The experimental scheme for investigation of intercellular contacts between T-cells and Jurkat cells infected by HIV is also proposed (McNerney et al. 2010). In this case, virological synapses playing a significant role in replication of viral particles in lymphocytes may be directly visualized.

Mixed models

The combination of advantages of abiotic and biotic models makes possible using of mixed models in experimental practice. Typically, the first component is a mineral carrier (microsphere or surface); the second one is a living cell (bacterial or eukaryotic). Eukaryotic cells are grown on a suitable horizontal plate, and a mineral carrier is approached horizontally (Fig. 4I). If a bacterial cell is used, the studied cell either can be static (adsorbed on a mineral surface or onto a carrier, e.g., the bead of larger diameter) or can be trapped in a laser focus. Described models may be two or three components—in the latter case, the bacterial cell is located between two carriers (Fig. 4G)—so that it is possible to measure the interaction force and to study surface structures of living or chemically treated cells (e.g., pili) (Fig. 4F–H).

Real-time studies of bacterial motility are of particular interest. In this case, the trapped microsphere is approached toward the bacterial cell adsorbed onto a solid surface (Fig. 4H). While bacterium gliding, the microsphere is displaced out of the focus center—this is registered as serial leaps on the chronogram of a signal (Tanaka et al. 2016).

Conclusion

Optical tweezers are the most common used biophysical instruments, along with atomic force microscopy, for studying molecular interactions by force spectroscopy. The theory of optical trapping and the physicochemical basis of short-distance interactions between colloidal surfaces in a liquid medium are covered in this review. The issues of using living cells as objects for trapping are briefly reviewed. Most modern models of optical tweezers allow registering forces in the range from 0.1 to 100 (200) pN, which makes it possible to study different biological structures on the surface of bacterial and eukaryotic cells (pili, filopodia, etc.) and also to evaluate ligand–receptor interactions. The several types of model systems based on the mineral carriers treated with target molecules or cells attached to surfaces are the subjects of considerable literature. They all can be conditionally divided into three types—biotic, abiotic, and mixed. The last two types are used most often due to their greater stability in time and to the fact that their components are less susceptible to the thermal effects of laser radiation. The use of an axial trap and a combination of optical trapping with Raman spectroscopy and fluorescence microscopy can be considered promising new directions.

Author contribution

Physical basis of optical trapping and calculation of the forces were summarized by Ilya Konyshev. Biological applications of laser tweezers were reviewed by Ilya Konyshev and Andrey Byvalov. The vector images were created with the participation of both authors.

Funding

This work was supported by the grant from the President of the Russian Federation for state support of Russian Candidates of Science (No. MK-3383.2021.1.4).

Declarations

Conflict of interest

The authors declare no competing interests.