Casimir Effect with Dielectric Matter in Salted Water and Implications at the Cell Scale

Casimir Effect with Dielectric Matter in Salted Water and Implications at the Cell Scale

Larissa Inácio, Centre of Excellence ENSEMBLE3, Warsaw, Poland. Felipe S. S. Rosa and Paulo A. Maia Neto, Instituto de Física, Universidade Federal do Rio de Janeiro, Brazil. Astrid Lambrecht, Forschungszentrum Jülich and RWTH Aachen University, Germany. Serge Reynaud, Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL, Collège de France, Paris.

Received 13 January 2026; revised 13 February 2026; accepted 28 February 2026; published 10 April 2026.

Abstract

The Casimir interaction in salted water contains a universal contribution of electromagnetic fluctuations that makes it of a longer range than previously thought. The universal contribution dominates non-universal ones at the distances relevant for actin fibers inside the cell. We discuss universal and non-universal contributions with a model mimicking biological matter. We also show that the universal Casimir effect should have crucial implications at the cell scale.

Keywords: electromagnetic fluctuations; Casimir effect; physics at the cell scale.

Displayed equations in this text are given as named results in words, with the article's own equation numbers; the complete algebra and the nine figures are at the source.

1. Introduction

Quantum physics was born when Max Planck wrote the first quantum law, where he explained the properties of black-body radiation. This law gave the mean energy per mode of the electromagnetic field as the product of the energy of a photon, the reduced Planck constant times the angular frequency, with a mean number of thermal photons per mode at that frequency. Approximately ten years later, Planck added an extra term of one half of the reduced Planck constant times the frequency to the mean energy per mode. This marked the appearance in physics of zero-point fluctuations, which have superseded the now abandoned reasoning of the paper in which they were introduced.

While the original Planck law fell off the expectation of the Boltzmann constant times temperature by a constant offset of minus one half of the photon energy at high temperatures, the second law including zero-point fluctuations had the correct classical limit. The modern writing of the mean energy per mode makes this property straightforward since all the terms in the high-temperature expansion go to zero at high temperature but the dominant one: equation (1) states that one half plus the mean thermal photon number, times the photon energy, equals half the photon energy times the hyperbolic cotangent of the photon energy over twice the thermal energy, which tends to the thermal energy itself in the high-temperature limit.

Discussions on zero-point fluctuations were active before their precise nature was better understood as the full development of quantum theory took place. Vacuum fluctuations, the zero-point fluctuations of electromagnetic fields in empty space, were introduced in previous quantum theory.

Nowadays, zero-point fluctuations are considered as a consequence of the non-commutative character of quantum observables. The dispersions of these fluctuations are specified by inequalities deduced from the commutators of observables. Vacuum fluctuations are now defined by the quantum theory of the electromagnetic field, with each field mode represented by canonical variables analogous to those of an harmonic oscillator. This theory leads to a full quantum derivation of Einstein's description of absorption, stimulated and spontaneous emission processes. Vacuum fluctuations have many essential effects in atomic and subatomic physics, such as the Lamb shift and radiative corrections.

In this paper, we focus the discussions on mechanical effects due to the radiation pressure of these fluctuations that are Casimir forces and related Casimir–Polder and van der Waals forces. In his initial study, Hendrik Casimir studied an idealized problem with two perfectly reflecting plane mirrors in an electromagnetic vacuum at zero temperature. Since then, a considerable number of papers have generalized the description to make it more realistic in terms of material properties and geometry. Meanwhile, more and more sophisticated experiments have, in particular, precisely measured the forces between metallic reflecting surfaces in empty space.

We stress at this point that most experiments are performed at room temperature, so thermal and zero-point fluctuations have to be considered together. The study of the Casimir effect in the presence of thermal fluctuations has a long history. It is conveniently addressed by using the Matsubara summation technique. The latter is directly related to equation (1), as Matsubara frequencies are nothing but the poles of the hyperbolic cotangent function: equation (2) writes the half-photon-energy hyperbolic cotangent as a sum over all integers of the thermal energy divided by the reduced Planck constant times the frequency offset from the corresponding imaginary Matsubara frequency, where the Matsubara frequencies are the integer multiples of two pi times the thermal energy divided by the reduced Planck constant.

An advantage of the Matsubara technique is that the high-temperature limit of the interaction is given by the term at zero frequency, that is, the pole at zero frequency in the formulas (1) and (2). This contribution has a universal character since it does not depend on details of the variation with frequency of the electromagnetic response functions.

For the most common experimental configuration using metallic scatterers in empty space, this universal limit can only be met at distances larger than the thermal wavelength in vacuum, where the force is quite small and hard to measure with precision. There exists, however, another configuration of great interest in this respect, which involves dielectric matter in electrolytes. In this case, the Casimir force can be both large and dominated by the universal term at distances that are not too large. This configuration has been studied in previous theoretical and experimental papers and is reviewed in the following.

This universal Casimir effect in electrolytes is of particular relevance at the interface of physics and biology as it may have quite a large impact on the mechanics of biological systems at the cell scale, a point that has been recently emphasized and is revisited below.

2. Scattering Theory of Casimir Interaction in Electrolytes

We first describe in this Section the scattering theory of the Casimir effect in electrolytes in the simplest geometry of two bulks of matter with parallel plane interfaces, separated by an electrolyte layer.

Figure 1. Sketch of the two-bulk configuration, with d the distance between the two parallel interfaces and the two dielectric functions for water and immersed matter.

As we consider the Casimir interaction between objects immersed in an electrolyte, the key "novelty" is that the dissolved ions give rise to essential nonlocal effects. Such nonlocality, also referred to as spatial dispersion, manifests itself in a wave number dependence of the permittivity in reciprocal space and also in the emergence of a nontrivial tensorial character of the electrolyte permittivity, equation (3): the scalar permittivity as a function of frequency is replaced by a permittivity tensor depending on both frequency and wave vector. The main consequence of the nonlocality of the medium is the support of longitudinal modes, in addition to the standard transverse electromagnetic fluctuations. The medium allows for three independent polarizations: transverse electric, transverse magnetic and longitudinal, and the extra degree of freedom for the fluctuations has considerable consequences for dispersion interactions.

The most economical way of writing the Casimir energy is by reinterpreting it as a reverberation problem for the electromagnetic fluctuations and then making a Wick rotation to imaginary frequencies. For simplicity, let us assume that the bulk materials are strictly spatially local. The Wick rotation produces a Matsubara sum over the frequencies defined in equation (2): equation (4) writes the total Casimir free energy as the zero-frequency term plus the sum of all the non-zero Matsubara terms. In the total Casimir free energy we separate the zero Matsubara frequency term, as it plays the main role in the discussions of this paper. The other terms, each corresponding to a non-zero Matsubara frequency, are discussed for the two-bulk geometry in the earlier literature, and they are discussed for the two-slab geometry in Section 5 below.

It can be shown that the transverse magnetic and longitudinal fluctuations are coupled at the interfaces between the bulks and medium, but, in effect, it turns out that the transverse electric term vanishes in the zero-frequency limit and the transverse magnetic and longitudinal contributions actually uncouple one from another. Equation (5) then gives the zero-frequency free energy per area as the sum of two integrals over transverse wave vector: the first, the transverse magnetic term, is the thermal energy over two times the logarithm of one minus the exponential of minus twice the wave number times the distance, which integrates to minus the thermal energy times Apéry's constant divided by sixteen pi times the square of the distance; the second, the longitudinal term, has the same form but with the reflection amplitude for longitudinal modes squared and with the wave number replaced by the square root of the wave number squared plus the inverse square of the Debye length.

Let us rewrite expression (5) by highlighting the first term, the transverse magnetic one, that is henceforth referred to as the universal Casimir contribution as it depends only on the temperature and geometry. Equation (6): the zero-frequency free energy is the universal bulk term plus the longitudinal term, where the universal bulk term is minus the area times the Hamaker constant divided by twelve pi times the square of the distance, and the Hamaker constant is three quarters of Apéry's constant times the thermal energy.

The Hamaker constant depends only on temperature — it is about 0.9 times the thermal energy, since Apéry's constant, the Riemann zeta function at argument three, is about 1.202 — and it yields the free energy through a multiplication by a dimensionless geometrical quantity. As the universal term is proportional to temperature, the associated entropy is deduced just to be such that the mechanical energy, the free energy plus the temperature times the entropy, vanishes, which means that the universal Casimir interaction is a purely entropic effect. These properties are still true for the other geometries considered below, with different expressions, naturally.

The second term in equation (5) is the longitudinal contribution at zero Matsubara frequency, and is determined by the reflection amplitude and the Debye length, equation (7). The reflection amplitude is the difference between the static permittivity of pure water times the square root of the wave number squared plus the inverse square Debye length and the static permittivity of the immersed matter times the wave number, divided by the sum of the same two quantities. The Debye length is the square root of the static permittivity of pure water times the thermal energy divided by the ionic concentration times the ion mass, where the pure-water permittivity means the permittivity of water without the ionic contribution.

In the long-distance limit, the longitudinal part is exponentially suppressed by the Debye screening with the typical Debye length: equation (8) states that it falls off as the exponential of minus twice the distance divided by the Debye length.

This is the first major takeaway of this paper: there is indeed a screening of the longitudinal-mode contribution across the electrolyte, but the universal contribution is unscreened as it arises from the transverse electromagnetic fluctuations. This property was first demonstrated in the framework of a macroscopic description of the dielectric response of salted water, and it has been confirmed since then by molecular dynamics simulations. This has far-reaching consequences for physical and even biological systems, which we discuss below.

An essential aspect has to be clarified at this point. The discussions just presented on transverse and longitudinal fluctuations treat fields in the vicinity of zero frequency rather than at zero frequency. In technical terms, what was calculated is given by a residue associated with the pole at zero frequency of the cotangent function in equation (2). The physical origin of the effect is merely the divergence, as the thermal energy divided by the photon energy, of the number of photons in a field mode with frequency close to zero, finally resulting in a finite contribution to the free energy.

3. Experimental Evidence with Optical Tweezers

Optical tweezers are considered a highly appropriate tool to measure interaction forces in aqueous media, and several applications in biology have been developed. Since the trap stiffness is proportional to the beam laser power, the system can be tuned to match the required order of magnitude. Standard single-beam traps are most effective for particles whose refractive indexes are slightly higher than that of the surrounding medium. Conveniently, this is also the condition that minimizes the non-universal contributions from the non-zero Matsubara frequencies, which depend on the reflection, or more generally scattering, amplitudes at the interface between the material and the host medium.

The universal Casimir interaction between two silica microspheres immersed in salted water has been measured with optical tweezers. Brownian fluctuations of a small silica microsphere, radius 2.4 micrometres, held by optical tweezers were measured along the two directions transverse to the laser beam. The interaction energy was then observed through the modification of the fluctuations along the axis connecting the sphere centres as the larger microsphere, radius 12 micrometres, adhered to the glass slide at the bottom of the sample, was approached by employing a piezoelectric nano-positioning system. The sample was also displaced vertically to align the sphere centres. While fluctuations along the connecting axis captured the interaction signal, fluctuations along the perpendicular axis were not modified, showing the absence of optical binding or of any other perturbation of the optical force in this configuration with near-index matching. Since the trapping beam was circularly polarized, the Brownian fluctuations along the perpendicular axis accumulated over several experimental runs were employed to infer the optical potential relevant for the interaction direction. As an additional check, the resulting optical potential was shown to agree with the Mie–Debye theory of optical tweezers as well as with standard Stokes calibration, with the latter implemented in the absence of the larger microsphere.

Figure 2. Principles of the experiment: a silica microsphere is held by a tightly focused laser beam close to a larger silica microsphere attached to the glass slide at the bottom of the sample chamber. The distance between the larger microsphere and the laser axis is controlled by using a piezoelectric nano-positioning system. The Brownian fluctuations of the trapped microsphere are measured for different values of that distance.

The interaction energy for distances above 100 nanometres was then obtained by subtracting the optical potential from the total energy as determined by the Brownian dynamics along the connecting direction. For moderate salt concentrations, the interaction was dominated by the electrostatic double-layer interaction. In order to suppress such interaction, a second experiment employed a high salt concentration, strong screening with a Debye length of 0.65 nanometres. In this second case, the interaction was completely dominated by the universal Casimir contribution discussed in the previous section.

Figure 3 shows the resulting Casimir energy in units of the thermal energy versus the distance of closest approach. The red curve represents the non-universal contribution accounting for the positive Matsubara frequencies. The calculation is based on Mie scattering by dielectric spheres in water. The red curve thus represents the generalization for the geometry of two spheres of the standard theoretical prediction for the Casimir interaction in salted water, which is often considered within the parallel planar interface setup or within the corresponding proximity-force approximation. The longitudinal contribution is not included as it is negligible at both moderate and high salt concentrations employed, due to the fact that the distances are much larger than the corresponding Debye screening lengths.

Figure 3. Variations in Casimir interaction energy, in units of the thermal energy, with the distance between two dielectric microspheres in salted water for the setup depicted in Figure 2. Experimental points are shown with their error bars. The old approach relying on overlooking the universal contribution leads to way too small values to be compatible with the experimental data. The new theory, with the universal contribution included, agrees fairly well with experiments, with no fitting.

The black curve in Figure 3 shows the total Casimir interaction, including the universal contribution arising from transverse magnetic modes in the zero-frequency limit. Like the non-universal contribution, the total Casimir interaction is calculated exactly for the geometry of two dielectric spheres in salted water. The comparison between the black and red curves indicates that the universal contribution dominates the Casimir interaction for distances above 200 nanometres. More importantly, the unscreened universal Casimir contribution provides an excellent description of the data, with no fitting parameter, in contrast with the standard approach that overlooks the contribution of transverse magnetic modes and is excluded by the experimental data shown in Figure 3.

4. Universal Casimir Interaction in the Two-Sphere or Two-Cylinder Geometries

The universal Casimir interaction that is the high-temperature limit of the expression in the case of highly efficient Debye screening has been calculated in the two-sphere and two-cylinder geometries. As in the two-bulk geometry considered above, it arises from the zero term in the Matsubara sum appearing due to transverse electromagnetic fluctuations, and it does not depend on details of the frequency dependence of dielectric functions.

The associated free energy is the product of the thermal energy scale by a dimensionless function, which now depends in a nontrivial manner on the geometrical parameters. In the two-sphere geometry, equation (9) gives the universal free energy as minus the thermal energy times a function of the distance of closest approach and the two sphere radii.

The Casimir interaction is captured in that function, which has been calculated for two dielectric spheres in salted water. The function does not depend on temperature — the effect is purely entropic — and it is a dimensionless function of the two ratios that can be formed with the three length parameters: the distance of closest approach and the radii of the two spheres.

The function is calculated within the scattering theory of the Casimir effect. It is a sum over all field modes of scattering operators involving an arbitrary number of round-trips in the cavity formed by the two objects. In the so-called dipolar limit where the spheres have small sizes in comparison with the distance, the contributions of large numbers of round-trips are negligible. In this small-size approximation, or equivalently long-distance approximation, equation (10) gives the function as three quarters of the product of the two sphere volumes, in the sense of the cubes of the radii, divided by the sixth power of distance.

In the opposite limit of a short distance between the spheres, the exact result involves large numbers of round-trips, and it tends to an expression related to the two-bulk geometry through the proximity-force approximation. In this limit, equation (11) gives the function as Apéry's constant times the effective radius divided by eight times the distance, where the effective radius is the product of the two radii divided by their sum.

The function has a universal expression that does not depend on the details of the variation with frequency of the permittivity functions of either the dielectric matter in the spheres or salted water constituting the immersion medium. The result is conveniently conveyed as a function of two dimensionless parameters, equation (12): an aspect ratio for the distance, the distance divided by the effective radius, and an aspect ratio for the radii, the product of the radii divided by the square of their sum. That the function depends mainly on the first of these is a remarkable scale-invariance property, meaning that it is unchanged when the three geometrical dimensions are multiplied by a common factor.

Figure 4. Geometry for two spheres or two cylinders immersed in salted water, with the radii of the spheres or cylinders and the distance of closest approach.

A more subtle property appears when the function is written in terms of another parameter, equation (13): the sum of the squared distance between centres and the two squared radii, divided by twice the product of the radii. The value of the function thus appears to depend mainly on this parameter, which reveals an approximate conformal invariance of the geometrical dependence of the free energy. This property has been used to provide a relatively simple formula that can be applied to a wide range of geometrical parameters. This formula is sufficient for the salinity of biological media and at ambient temperature. The obtained free energy is larger than the thermal energy only for spheres quite close one to another, that is, the geometry in which the universal Casimir interaction has been experimentally measured.

In order to compare the two-sphere case and the two-cylinder one discussed below, we consider the geometry of two spheres with equal radii and show the corresponding function against the distance aspect ratio. The greyed-out band in the figure shows where the function is less than one, which is also where the binding energy is smaller than the Brownian motion energy scale. The figure shows that the universal Casimir effect dominates the Brownian energy for distances below about 0.09 of the effective radius. Note that, for the geometry with different radii employed in the optical tweezers experiment of Section 3, the order of magnitude for the border where the universal free energy equals the thermal energy is at about 0.2 micrometres, which is indeed the scale of distances probed in the experiment.

We now come to the discussion of the configuration with two dielectric cylinders in salted water. This configuration makes the interaction larger than in the two-sphere case as it scales as the length of the cylinders. The geometry is two cylinders with two radii at a distance of closest approach. In this geometry, equation (14) gives the universal Casimir free energy as minus the thermal energy times the ratio of cylinder length to distance times a second dimensionless function of the distance and the two radii.

The physics of the universal interaction is captured in that second function. The remarks just after equation (9) remain valid here as soon as the function does not depend on temperature and is a dimensionless function of two ratios of the three geometrical quantities.

In order to discuss the implications of these results for biological systems in Section 6, we depict the function for two cylinders with equal radii, plotted against the same distance aspect ratio. The greyed-out band shows where the function is less than one thousandth, that is, where the binding energy is smaller than the Brownian motion energy scale for a length one thousand times the distance. The key message is that the universal Casimir free energy dominates the Brownian motion energy imposed by the surrounding water over a broad range of parameters, including in particular distances of the order of the effective radius. The striking difference between the two-sphere case and the two-cylinder one is just due to the presence of the considerably large length-to-distance factor in equation (14), which makes the cylinder free energy larger than the thermal energy in a wider range of parameters.

Figure 5. Universal functions showing the variations in the universal free energy against the ratio of distance to effective radius: the function calculated for two spheres with equal radii, and the function calculated for two cylinders with equal radii. Dotted and dashed lines represent the proximity-force and small-size approximations that provide the asymptotic behaviours at small and large values of the ratio, respectively.

Both functions in Figure 5 agree with the proximity-force approximation at short separations and with the small-size approximation at long distances. For completeness, equation (15) gives the two approximations for the case of two cylinders: in the proximity-force limit the function is Apéry's constant over thirty-two times the square root of twice the effective radius divided by the distance; in the small-size limit it is 891 pi over 4096 times the product of the squared radii divided by the fourth power of the distance.

In particular, the power-law changes have clear meanings. For the proximity-force approximation, the cylinder function is proportional to the square root of effective radius over distance because there is only one curvature radius in a direction orthogonal to the distance for cylinders, whereas there are two of them for spheres, and the sphere function is proportional to the effective radius over distance. For the small-size approximation, the extensive quantity in the perturbative limit is the area for cylinders rather than the volume for spheres. It follows that the cylinder function is proportional to the two areas, the squared radii.

5. Non-Universal Contributions with Dielectric Matter in Salted Water

The universal Casimir interaction was calculated for two dielectric cylinders in salted water, but the non-universal contributions have never been analyzed in this geometry. In order to fill this gap, one needs a robust model for the dielectric response of matter as non-universal contributions depend on this response. In the absence of an exact solution, we have also to produce a reasonably accurate estimate for the interaction between two cylinders. These requirements are tackled in this Section.

For describing dielectric properties of organic matter, we follow earlier work by considering tetradecane as a proxy for these properties. Its dielectric function is given by a Lorentz model. In addition, we modelled salted water with the contribution from the ions depending on their concentration. As we mentioned in Section 4 above, this contribution is critical for the universal contribution at zero frequency, but it plays a negligible role at non-zero Matsubara frequencies, which are far too high at room temperature to "feel" the effect of ionic conduction — compared to electrons, ions have a much larger mass and their number per unit volume is much smaller.

Figure 6. Dielectric functions of water and of tetradecane against imaginary frequency. The vertical dashed line represents the first non-zero Matsubara frequency at ambient temperature, where the dielectric functions of water and tetradecane are nearly identical.

The tetradecane function is represented in Figure 6 against imaginary frequency, that is after a Wick rotation as is common for studies of the Casimir effect, alongside the pure water function. The vertical dashed line shows the position of the first non-zero Matsubara frequency at room temperature. The figure indicates a nearly perfect index matching of biological matter and water at this frequency, which is at the root of the smallness of the non-universal contributions to the Casimir interaction. Although the contribution of ions is negligible at this frequency, the index contrast grows sharply for lower frequencies, in particular as the effective response diverges for salted water. This Drude-like divergence caused by the dissolved ions is the reason why the universal contribution ends up not depending on the details of the properties of the material immersed in salted water.

In order to get reliable estimations for the non-universal contributions to interaction between cylinders, we use the fact that these contributions are considered to be comparably small due to the index-matching at most non-null Matsubara frequencies and, therefore, that they should be approximately represented by a pairwise summation technique. Let us stress that such an approximation cannot give an accurate description of the universal contribution as perturbation theory is in no way adapted to this case. Guided by the pairwise summation, we consider that the order of magnitude of the ratio between universal and non-universal contributions arising from two cylinders of a given radius can be approximately estimated from the same calculation in a setup of two slabs of thickness equal to the diameter, separated by a gap of salted water.

Figure 7. Sketch of the two-slab configuration, with the distance between the two parallel interfaces and the width of the slabs.

Rewriting expression (4) for slabs, equation (16) again separates the zero Matsubara term from the sum of the non-zero ones. Equation (17) gives each non-zero Matsubara term as the area times the thermal energy times a sum over the two polarizations of an integral over transverse wave vector of the logarithm of one minus the squared slab reflection coefficient times the exponential of minus twice the longitudinal wave number in the medium times the distance. Equation (18) gives the reflection coefficients of a slab in terms of the Fresnel coefficient and the slab thickness — the Fresnel coefficient times one minus the exponential of minus twice the longitudinal wave number in the slab times the thickness, divided by one minus the squared Fresnel coefficient times that same exponential — with the longitudinal wave numbers being the square roots of the corresponding permittivity times the squared Matsubara frequency over the squared speed of light plus the squared transverse wave number. Equation (19) gives the Fresnel coefficients for a plane wave going from the electrolyte to the slab, the standard transverse electric and transverse magnetic expressions in those two wave numbers and the two permittivities.

The zero-frequency Matsubara term for the slabs is analogous to expression (5). Given that we are interested in distances much larger than the Debye length, the longitudinal part is negligible and equation (20) states that the zero-frequency slab term equals the universal slab term. The universal contribution for two slabs is the same as in the case of two half-spaces discussed in Section 2.

The variations in the universal contribution to the binding energy and in the non-universal part against distance are depicted separately in Figure 8, for identical slabs of thickness six nanometres, a number motivated by the discussions in Section 6. The non-universal contributions have been calculated with the dielectric model for tetradecane represented in Figure 6. Figure 8 clearly shows that the non-universal contributions are much smaller than the universal one, which is a consequence of the index-matching at non-zero Matsubara frequencies between water and tetradecane. We have checked that the ratio between the two contributions is always above a factor 100.

Figure 8. Universal and non-universal contributions to the Casimir binding energy for two slabs of tetradecane of thickness six nanometres separated by a salted water gap. The binding energy per unit area is represented in units of thermal energy per square micrometre.

6. Implications at the Cell Scale

We now use the results obtained to show how the universal Casimir interaction may play an essential role in biomechanics at the cell scale. Here, we only recall a few critical points.

Actin filaments play critical roles in the mechanical functions of the cytoskeleton, as well as in many intracellular processes, by actively generating forces with the help of motor proteins. The actin filaments form bundles, with filaments cross-linked by specific proteins into parallel arrays. It has been shown that bundles of parallel filaments may form in vitro in the absence of cross-linkers. Other examples of self-assembled filaments play key roles in or beyond the cytoskeleton. The universal Casimir interaction considered in this paper is relevant at the observed dimensions for bundles of actin filaments, so it should have essential implications in the self-assembly and cohesion of bundles of filaments at the cell scale.

Estimation of the interaction for the relevant case of parallel actin filaments is conducted by modeling filaments as cylinders and using the physical dimensions known for parallel actin filaments, each having a radius of about 3 nanometres and a length of about 15 micrometres, which form bundles with a typical distance of closest approach of about 6 nanometres between filaments. Such distance is about ten times larger than the characteristic Debye screening length in biologically relevant solutions. The electrostatic interaction as well as the longitudinal contribution to the Casimir interaction are then negligible as they are efficiently screened. The parameters given above correspond to a large value for the length-to-distance ratio, about 2500, which is a key reason that the corresponding Casimir interaction is as significant. The binding energy, expressed in units of the thermal energy, is shown against the separation distance in Figure 9. The relevant distance, about 6 nanometres, is emphasized as the red point, where the binding energy is of the order of five times the thermal energy. In the grey-shaded zone, the binding energy is dominated by the Brownian motion energy.

Figure 9. Casimir binding free energy between actin filaments, each having a radius of 3 nanometres and a length of 15 micrometres. The binding free energy is drawn against the distance between two parallel actin filaments and measured in units of the thermal energy. In the grey-shaded zone, it is dominated by the thermal energy. Energies above this zone are relevant for biomechanics in the cell, and they include the physiological distance of 6 nanometres indicated by the red solid dot. Inset: schematic of two actin filaments in a two-filament bundle, with cross-linkers shown in red.

Let us stress at this point that such a magnitude had to be expected for a physical interaction having relevance for biomechanics at the cell scale. The interaction has to be larger than the thermal energy in order to dominate Brownian agitation, while it should not be so large as to be incompatible with the mechanical activity provided by molecular motors. This was underlined by Moysés Nussenzveig in his remarkable paper commenting on the links between physics and biology, and discussing in particular the pioneering studies by Niels Bohr and Erwin Schrödinger.

We also emphasize that this magnitude has been obtained without any ad hoc fine-tuning of the parameters. The energy constants are given by the theory of the universal Casimir interaction reviewed in this paper, and they do not depend on any specific detail of the biological matter involved. Hence, the binding energy depends only on geometrical parameters known for bundles of actin filaments. Note also that the numbers presented for actin correspond to a distance aspect ratio of four in Figure 5b, where the proximity-force expression overestimates the correct value by a factor of 13. This implies that the full cylinder calculation was mandatory for obtaining a reliable estimation for bundles of actin filaments.

Another crucial point becomes clearer after the discussions in the present paper. We gave here an estimation of the non-universal Casimir interaction in Section 5, showing that it is more than a hundred times smaller than the universal one. For the relevant distance of 6 nanometres between actin filaments, the non-universal contribution is then smaller than 0.05 times the thermal energy. It is thus established that the non-universal interaction is not only much smaller than the universal one but also way too small to have any relevance in a medium dominated by Brownian agitation. It is only the presence of the universal contribution that makes the Casimir binding energy essential for biomechanics at the cell scale.

As a concluding remark, let us stress that physics is highly complex at the cell scale, with quite a large number of different processes involved in the mechanics and functionalities of the cell. Our modest aim in this review has been to show that the universal Casimir interaction due to transverse electromagnetic fluctuations has to be considered as a significant contribution to this rich physics as it produces an attractive interaction between biological filaments in the cell with a typical energy of a few Brownian thermal scales.

(Author contributions, funding, acknowledgements and the reference list of the published article are omitted here; the complete text is at the source.)