Universal Casimir attraction between filaments at the cell scale

Abstract

The electromagnetic Casimir interaction between dielectric objects immersed in salted water includes a universal contribution that is not screened by the solvent and therefore long-ranged. Here, we study the geometry of two parallel dielectric cylinders. We derive the Casimir free energy by using the scattering method. We show that its magnitude largely exceeds the thermal energy scale for a large parameter range. This includes length scales relevant for actin filaments and microtubules in cells. We show that the Casimir free energy is a universal function of the geometry, independent of the dielectric response functions of the cylinders, at all distances of biological interest. While multiple interactions exist between filaments in cells, this universal attractive interaction should have an important role in the cohesion of bundles of parallel filaments.

1. Introduction

The electromagnetic Casimir or van der Waals attraction between dielectric particles immersed in salted water was recently shown to be stronger and of longer range than previously expected. This long-range Casimir interaction was predicted as an effect of non-screened electromagnetic thermal fluctuations confined between plane dielectric surfaces. For spherical particles, the interaction is a universal function of distance, independent of the dielectric response functions of the particles, and it overtakes non-universal contributions at distances of the order of 0.1 micrometres. On the other hand, such non-universal contributions dominate the total interaction when probing the force between dielectric spheres at distances in the nanometer range. The existence of the non-screened universal Casimir force was proven experimentally on a microsphere held by optical tweezers interacting with a larger rigidly held sphere at distances above 0.2 micrometres. A long-ranged attraction at similar distances was also found for optically trapped dielectric microspheres in salted water.

Between spherical particles, this universal Casimir interaction only dominates the thermal energy scale associated to Brownian motion in the liquid when the distance between spheres is smaller than one tenth of the smallest radius. However, there are other highly relevant geometries where this interaction should be more significant. Here, we study the case of two parallel dielectric cylinders in salted water, where the force is expected to be proportional to the length of the cylinders, itself much larger than the radial dimensions. We show that the electromagnetic Casimir attraction in such configuration can indeed dominate the thermal energy at distances larger than the radii.

Considering dielectric cylinders immersed in salted water allows us to address the following question: Can the universal Casimir interaction play an important role in biological systems at the cell scale? Indeed, filamentous structures are ubiquitous in cells. Cytoskeletal filaments, in particular actin filaments and microtubules, play crucial parts in maintaining the integrity of eukaryotic cell shape, in its deformations, as well as in multiple sub-cellular processes, by actively generating forces with the help of motor proteins. Actin filaments form bundles, where filaments are cross-linked by specific proteins into parallel arrays. Microtubules, which are thicker and more rigid than actin filaments, also form bundles cross-linked by microtubule-associated proteins. Both in the case of actin filaments and in that of microtubules, bundles of parallel filaments have been shown to form in vitro in the absence of cross-linkers under certain experimental conditions. Beyond the cytoskeleton, several enzymes form filaments in cells, with important biological functions, and these filaments also often self-assemble into larger assemblies, especially bundles. The Casimir interaction considered in this paper matters in particular at dimensions relevant for bundles of actin filaments and of microtubules. Therefore, it should have important implications in the self-assembly and cohesion of bundles of filaments at the cell scale.

Let us note that the configuration with metallic cylindrical surfaces separated by a vacuum gap was proposed as a platform for precision Casimir experiments. The Casimir force between crossed cylindrical surfaces in air was measured for distances up to 0.1 micrometres. Theoretical results for metallic cylindrical surfaces in vacuum were derived at zero and finite temperatures, as well as in a non-equilibrium configuration.

This work is organized as follows. First, we illustrate by molecular dynamics simulations the fact that transverse modes of the electromagnetic field are not screened by ions, which is crucial to the existence of a long-range Casimir force. Then, we calculate the Casimir interaction between two parallel dielectric cylinders immersed in salted water, using the scattering formalism, and we discuss its universality. Next, we apply our results to bundles of biological filaments, focusing on the specific cases of actin filaments and microtubules. Finally, we discuss the quantitative importance of the electromagnetic Casimir attraction in these biological systems.

2. Transverse electromagnetic modes are not screened

Despite strong screening, the Casimir interaction includes a long-range unscreened part, due to the effect of thermal electrodynamical fluctuations propagating in the medium without being screened. To illustrate this key point, while relying on a molecular description of the environment, we perform molecular dynamics simulations, supported by a classical field theory calculation. We simulate pure water using a classical rigid model for water molecules, as well as an electrolyte solution with concentration 0.2 mole per liter of potassium bromide. This is in the range of typical cytoplasmic concentrations, and is thus relevant for our applications to bundles of biological filaments below.

We compute the static dielectric correlation spectrum in Fourier space for these two media. Longitudinal and transverse correlation functions are expressed from the spatial distribution of the charges in the medium and averaged on the simulation time. Looking first at the longitudinal susceptibility for pure water and for the electrolyte, over wavevector norms up to 1.5 per ångström since we focus on long-range interactions, we observe that the longitudinal susceptibility of the electrolyte significantly differs from the pure water one at low wavevector. Looking then at the transverse susceptibility for water and for electrolyte over the same range, we observe that the transverse susceptibility is not affected by the salt, in agreement with a previous study.

In addition, we use classical field theory to compute the response functions of electrolytes. We express the longitudinal and transverse response of pure water using the framework of nonlocal electrostatics. We adjust the two parameters of the model to fit the data of molecular dynamics. In water, for this range of wavevector, the longitudinal susceptibility is constant whereas the transverse one presents a Lorentzian decay. The transverse susceptibility of the electrolyte is unchanged when compared to pure water. Conversely, the longitudinal one presents a Lorentzian decay induced by Debye screening in the electrolyte. A very good agreement is obtained between field theory and simulations.

We checked the robustness of these conclusions by simulating an aqueous electrolyte using another water model and sodium chloride ions instead of potassium bromide. We indeed obtained similar results.

We thereby confirm that the longitudinal spectrum is modified by the presence of salt, due to the screening of the correlations in electrolytes beyond the Debye length, whereas the transverse correlation spectrum remains unaffected by the presence of salt. The absence of screening of the transverse modes by salt allows the long-range Casimir force.

3. Casimir interaction between two dielectric cylinders

We use the scattering formalism to calculate the Casimir interaction in salted water at room temperature between two parallel dielectric cylinders with length L and different radii, separated by a distance d of closest approach. The cylinder-plane configuration is included in the calculation, for an infinite second radius. We focus on cylinders much longer than the separation distance, thus neglecting edge effects. We consider a salt concentration typical of biological media, with the Debye screening length much smaller than the distance d. All electrostatic interactions, as well as contributions to the Casimir energy arising from longitudinal modes, are then efficiently screened. This is a first reason why the resulting interaction will be independent of many details of the physical configuration. This universal Casimir interaction arises from transverse modes, which are not screened, as discussed in the previous section.

Another reason for this universality will become clear when describing the scattering formalism employed to compute the Casimir interaction for arbitrary values of the geometrical dimensions. In general, the interaction is given by a sum over Matsubara frequencies. The first Matsubara term overtakes all other ones when thermal fluctuations dominate, which is the case for filaments at the cell scale at physiological temperature. This first term corresponds to electromagnetic response functions evaluated at zero frequency. As salted water features an ionic conductivity leading to a divergence of its contribution to the dielectric response, the resulting interaction does not depend on the detailed dielectric function of the cylinders.

Within the scattering approach, the zero-frequency term giving the thermal Casimir interaction energy is written, in equation (1), as the thermal energy scale times the cylinder length times an integral over the wavevector along the axis of the cylinders of the logarithm of the determinant of the identity minus the round-trip operator, divided by two pi.

Equation (2) defines the round-trip operator after scattering on the two cylinders as the product of the reflection operator on cylinder one, the translation operator from cylinder two to cylinder one, the reflection operator on cylinder two, and the translation operator from cylinder one to cylinder two.

The round-trip operator can be written in terms of cylindrical modes associated to given values of the axial wavevector and of an integer angular momentum component along the symmetry axis for each cylinder. The zero-frequency term giving the thermal Casimir interaction is calculated at the static limit for all reflection operators. As a consequence of the Debye screening mechanism, only the transverse-magnetic cylindrical modes contribute, that is, modes with the magnetic field perpendicular to the symmetry axis.

Due to the rotational symmetry of each cylinder, the reflection operators are diagonal in the representation defined by the cylindrical modes. The corresponding matrix elements are evaluated by taking into account the finite conductivity of salted water due to the ions in solution. As the dielectric permittivity of salted water diverges in the limit of zero frequency, the matrix elements do not depend on the dielectric response of the cylinder material. Using the known reflection matrix for cylinders, we derive in equation (3) that each diagonal transverse-magnetic reflection matrix element is minus one raised to the angular momentum index, times i pi over two, times the ratio of the derivative of the modified Bessel function of the first kind to the derivative of the modified Bessel function of the second kind, both evaluated at the axial wavevector times that cylinder radius.

The distance between the axes of the two cylinders is the separation of closest approach plus the two radii. Translations along the axis joining them are described by matrix elements, given in equation (4), proportional to the modified Bessel function of the second kind of the difference of the two angular momentum indices evaluated at the axial wavevector times the axis-to-axis distance, with a phase factor of plus or minus i raised to that index difference and a prefactor of minus two i over pi. Here we used Graf’s addition theorem for Bessel functions.

Explicit results for the Casimir free energy are obtained by combining these equations. We write the result, in equation (5), as minus the thermal energy scale times the ratio of the cylinder length to the separation distance, times a dimensionless function of the three radial dimensions: the separation of closest approach and the two radii. The free energy is thus proportional to the cylinder length, with the latter measured as the dimensionless ratio of length to separation. The dimensionless quantity depends only on the two ratios of these three dimensions. Thus, the free energy does not depend on any material properties of the cylinders or of the surrounding fluid, and is universal.

A convenient representation, given in equation (6), writes that dimensionless function in terms of two parameters. The first is a symmetrized ratio of the two radii: the product of the radii divided by the square of their sum. It runs from zero in the cylinder-plane geometry to one quarter for equal radii. The second compares the distance of closest approach to an effective radius, defined as the product of the radii divided by their sum; the effective radius equals the cylinder radius in the cylinder-plane geometry, and half the radius for cylinders of equal radii.

Plotting that dimensionless function for four values of the radius-ratio parameter, the Casimir interaction energy per unit length is obtained by multiplying it by minus the thermal energy scale divided by the separation; the free energy is negative and the dimensionless function positive. We optimized the numerical evaluation by expanding the round-trip operator in the plane wave basis rather than in the cylindrical one. Explicit expressions for the scattering matrix elements in the plane wave basis are readily derived from the results presented above. Since the wave-vector is a continuous variable, the determinant is calculated with the help of Nyström discretization, as in the calculation of the Casimir interaction between spheres. To facilitate applications, we provide the numerical evaluations on a repository for the four values of the radius-ratio parameter shown, over the range of reduced distances from 0.1 to 15, which should be appropriate for most applications.

Our full numerical results hold for arbitrary values of distance. Let us compare them with the proximity force approximation, or Derjaguin approximation, in the limit of small reduced distance. This approximation amounts to replacing the dimensionless function by the result for parallel planes averaged over the local distances between the cylindrical surfaces, yielding equation (7): the effective Hamaker coefficient divided by 24, times the square root of two over the cube of the reduced distance, where the Hamaker coefficient is three quarters of Apéry’s constant times the thermal energy scale. That effective Hamaker coefficient is approximately nine tenths of the thermal energy scale, the value calculated for dielectric surfaces separated by salted water, with Apéry’s constant approximately 1.202. The proximity force approximation is indeed a good approximation of numerical results at short distances, whereas it increasingly overestimates the magnitude of the interaction energy as the reduced distance grows. Note that this expression does not depend on the radius-ratio parameter.

Analytical results can also be derived in the opposite limit of large reduced distance. Indeed, the single round-trip approximation — replacing the logarithm of the determinant by minus the trace of the round-trip operator — is then sufficient to get an estimate of the free energy. We thus find, in equation (8), two different results for the case of two cylinders and for the cylinder-plane geometry: for two cylinders the dimensionless function falls as 891 pi divided by 4096, over the square of the radius-ratio parameter and the fourth power of the reduced distance; for the cylinder-plane geometry it falls as 7 over 64 times the square of the reduced distance. The comparison between these two long-distance results indicates that the reduction with respect to the proximity force limit is stronger for two cylinders than for a cylinder and a plane, as expected.

A key feature of the results obtained here is their universality. Our results are valid for whatever dielectric response functions of the cylinders. They depend only on the dimensionless length-to-distance ratio and on the dimensionless ratios of the radial dimensions. Furthermore, we show in an appendix that most of the dependence on radial dimensions is captured by considering the dimensionless free energy as a function of a conformally invariant geometrical parameter.

4. Application to bundles of biological filaments

How relevant is the universal Casimir attraction between dielectric cylinders in biological systems? To address this question, it is important to compare the magnitude of the Casimir free energy to the thermal energy scale. For actin filament bundles and for microtubule bundles, the filament length will be in the micrometer range while the inter-filament distance will be in the nanometer range. Thus, typically, the ratio of cylinder length over separation distance is of the order of one thousand for bundles in cells. With such a value, finite-size or edge effects are expected to be negligible, which is consistent with our assumptions. Importantly, we find that the Casimir binding energy is larger or of the same order as the thermal scale in a broad range, namely for reduced distances up to about five. As expected from the fact that the Casimir energy is proportional to the length of the cylinders, the range where the Casimir force plays an important role is much broader in the two-cylinder geometry, reduced distances up to about five, than in the two-sphere geometry, where it was evaluated as reduced distances up to about a tenth.

The values of reduced distance corresponding to the case of the filament bundles in cells discussed below are such that the Casimir force should play an important role in these systems. In addition, these practically relevant values also lie right in the crossover between the proximity force and the long-distance limits. In this intermediate range, both short- and long-distance approximations overestimate the exact energy that we computed numerically by approximately one order of magnitude. This highlights the importance of our calculation and of our full numerical results for these applications. The results obtained here are therefore of importance for the self-assembly and cohesion of filament bundles in cells, with implications for cellular and molecular biology.

Let us now assess more precisely the magnitude of this interaction in the specific case of actin bundles. Actin filaments are double helices of homopolymers of monomeric actin. They can be approximately described as cylinders with a radius around 3 nm. They form bundles in cells, where actin filaments are cross-linked by specific proteins into arrays of parallel filaments. In parallel actin bundles, which support projections of the cell membrane such as microvilli, microspikes or filopodia, actin filaments assembled with fimbrin, fascin or villin are approximately 6 nm apart, using the closest approach distance here and throughout. In this case, the free-energy relation yields a Casimir binding free energy per unit length of 0.33 thermal energy units per micrometre, giving the substantial value of 5 thermal energy units for a length of 15 micrometres, which is on the order of the size of a cell and of the persistence length of actin filaments. Such a value, significantly larger than the scale of thermal fluctuations, demonstrates the practical relevance of the Casimir interaction between actin filaments in the physiological configuration of parallel bundles. In contractile bundles, which are present in stress fibers, and in the mitotic contractile ring, actin filaments assembled with alpha-actinin are separated by 33 nm, yielding a smaller value of about one hundredth of a thermal energy unit for a length of 15 micrometres, which is not relevant as it is well below the scale of thermal fluctuations. Plotting the Casimir binding free energy against separation for actin filaments 15 micrometres long, the Casimir interaction exceeds the scale of thermal fluctuations for separations up to about 10 nm, which includes parallel bundles but not contractile bundles.

Another biologically important system where Casimir interactions between filaments are relevant regards microtubule bundles. Microtubules can be viewed as cylinders with a radius of about 12 nm. They can grow as long as 50 micrometres and their persistence length is around 1 mm. They form bundles where the separation between neighboring microtubules is set by microtubule-associated proteins, of which various types exist. Plant cells often possess large arrays of parallel microtubules, whose alignment is maintained over the whole cell, and where separations are similar to the microtubule diameter. Microtubule bundles are also present in neurons, where they play important roles, and the separations between adjacent microtubules in Purkinje cell dendrites, parallel fiber axons and white matter spinal cord axons were found to be 64 plus or minus 10 nm, 22 plus or minus 10 nm and 26 plus or minus 10 nm, respectively. For a separation of 22 nm, we find a binding free energy per unit length of about 0.112 thermal energy units per micrometre, giving 5.6 thermal energy units for a length of 50 micrometres and 1.7 thermal energy units for a length of 15 micrometres. Plotting the Casimir binding free energy against separation for microtubules 50 micrometres long, the Casimir interaction exceeds the scale of thermal fluctuations for separations up to about 35 nm, which includes the physiological separations found in parallel fiber axons and white matter spinal cord axons, as well as in plant cells, but not in Purkinje cell dendrites.

5. Discussion

Actin bundles and microtubule bundles are typically held in place by cross-linking proteins in cells. However, in electrolyte solutions containing polycations, for example magnesium ions, actin filaments can form bundles in vitro in the absence of cross-linking proteins. Microtubule bundles can also self-assemble in vitro above a certain concentration of multivalent cations. This demonstrates that the tendency of these filaments to self-assemble is quite generic. In this light, cross-linkers could help maintain spacing between filaments. Interestingly, beyond cytoskeletal proteins, multiple enzymes form filamentous structures within cells, which then assemble into large-scale self-assembled structures — foci, rods, rings, sometimes called cytoophidia — which are membraneless and reversible. Enzyme filamentation is associated to multiple functions in cells, including determining cell shape or regulating enzyme activity. Enzymes that form filaments and higher-order structures in cells include acetyl CoA carboxylase, CTP synthetase, inositol monophosphate dehydrogenase, and many others. Beyond enzymes, it was recently shown that mutating proteins that spontaneously form symmetric homo-oligomers can lead to polymerization in a quite generic manner. Furthermore, these mutant proteins very often form larger structures such as fibers or foci, and some were shown to bundle. These findings hint at a general trend of filaments to self-assemble into higher-order structures in cells.

While the universal attractive interaction discussed here should play a key role in these various bundles, these systems are complex and involve many other interactions. Electrostatic interactions are screened at the usual separations involved in actin bundles and microtubule bundles, but they matter at shorter separations. For instance, the surface of filamentous actin is overall negative, with a highly heterogeneous charge distribution, leading to subtle collective dynamics of counterions close to actin filaments. In addition, the depletion interaction, which arises from excluded volume effects on crowding agents, is important in cells due to how crowded the cytoplasm is. Indeed, around 30 per cent of its volume is estimated to be occupied by macromolecules, with notable heterogeneities. The range of the depletion interaction is given by the diameter of a typical depletant, which corresponds to macromolecules such as globular proteins in cells, with diameters of order 5 nm. Depletion interactions have been studied experimentally in controlled in vitro systems where the concentration of depletant polymers can be tuned. For instance, an adhesion strength of 7 thermal energy units per micrometre was measured between sickle hemoglobin fibers in a solution of monomeric hemoglobin, while the attractive interaction between two actin filaments in a solution of depletant polymers was found to be of order of a few times 10 thermal energy units per micrometre, and a similar value was found between two microtubules. Thus, this interaction is strong between parallel filaments in a cell, but it is also very short-ranged, with a range of order 5 nm.

What sets the Casimir interaction we calculated apart from electrostatic and depletion interactions, and to our knowledge, from all other interactions at equilibrium, is its long range, which arises from the lack of screening of transverse electromagnetic fluctuations. An out-of-equilibrium long-range fluctuation-induced interaction was recently predicted between neutral objects immersed in electrolytes subject to an external electric field. This force can be of importance at the cell scale, for example for ion channels. More generally, Casimir forces present interesting out-of-equilibrium properties. Such effects could be all the more important for cytoskeletal filaments in that the cytoskeleton is an active system. Here, we showed that an equilibrium long-range universal interaction exists between filaments in cells.

The Casimir interaction is highly dependent on the geometry of the interacting objects because it arises from the perturbation of electromagnetic fluctuations by the interacting objects. Here, we showed that its magnitude is several times the scale of thermal fluctuations in the geometry of two parallel filaments whose length, at the micrometer scale, is much larger than their radius and separation, at the nanometer scale. This is stronger than between two spheres, because long cylinders have a stronger confining effect on electromagnetic fluctuations than spheres. Accordingly, within a cell, the Casimir interaction can be strong between long semi-rigid biopolymers such as those considered here, but is weak between globular proteins. Note that similar geometric effects exist for other fluctuation-induced interactions, for example in the case of Casimir-like interactions induced by the thermal fluctuations of the shape of a biological membrane: these interactions are stronger between long parallel rods adsorbed on a membrane than between circular or point-like inclusions modeling transmembrane proteins. Critical Casimir forces can also be important for cylindrical particles immersed in critical binary mixtures. In addition to parallel cylinders, another biologically relevant case where the electromagnetic Casimir interaction should matter regards stacks of lipid membranes. Modeling them by parallel dielectric planes immersed in salted water, the Casimir binding free energy between two lipid membranes is 2.4 hundredths of a thermal energy unit times the area of the planes divided by the square of their separation. It should thus oppose the repulsive Helfrich and hydration interactions.

6. Summary and conclusion

The long-range part of the Casimir attraction has a universal form between two dielectric objects immersed in salted water. Here, we calculated the Casimir interaction in the case of two long parallel dielectric cylinders, using the scattering formalism. We demonstrated that this interaction takes values substantially larger than the scale of thermal fluctuations, in the important biological cases of actin bundles and microtubule bundles. The long range of the Casimir interaction we calculated arises from the lack of screening of transverse electromagnetic fluctuations, which we confirmed by molecular dynamics simulations. It is this long range that makes the Casimir interaction quantitatively important, for example between actin filaments at the physiological separation found in parallel bundles. It also sets it apart from other equilibrium interactions present in these structures. The Casimir interaction should thus play an important part in the self-assembly of filament bundles in cells.

Data availability statement

The data that support the findings of this study are openly available at https://doi.org/10.5281/zenodo.7634525. Code for our numerical calculations of the Casimir force is freely available in the GitHub repository https://github.com/sprengjamin/CasCy.

(Figures, appendices A to C and the reference list are omitted for length; the complete text is at the source.)