A new and computationally efficient version of the immersed boundary method, which is combined with the coarse-graining method, is introduced for modeling inextensible filaments immersed in low-Reynolds number flows. This is used to represent actin biopolymers, which are constituent elements of the cytoskeleton, a complex network-like structure that plays a fundamental role in shape morphology. An extension of the traditional immersed boundary method to include a stochastic stress tensor is also proposed in order to model the thermal fluctuations in the fluid at smaller scales. By way of validation, the response of a single, massless, inextensible semiflexible filament immersed in a thermally fluctuating fluid is obtained using the suggested numerical scheme and the resulting time-averaged contraction of the filament is compared to the theoretical value obtained from the worm-like chain model.

Living cells display a high degree of internal mechanical and functional organization and their intracellular biopolymeric scaffold, the cytoskeleton, plays a key role in that [

Actin filaments are biopolymers with sufficient contour length to exhibit significant thermal bending fluctuations, in the order of approximately 1% of their contour length. However, their diameter can be as large as ten nanometers or more, giving them noteworthy bending rigidity. Thus, actin filaments are said to be semiflexible in the sense that their bending stiffness is large enough for the bending energetics—which favors a straight conformation—to just out-compete the entropic tendency of a chain to crumple up into a random coil [

There has been deep interest in studying the mechanical response of biological tissues the past decades, and more specifically, in understanding the mechanical properties of biopolymeric networks, since they play an important role in cell motility [

Yamamoto et al. [

In the preceding studies the inextensibility constraint is not enforced strongly and this may lead to numerical errors [

In this study, a new computationally efficient version of the IBM, which is combined with the CGM, is introduced in this study for modeling inextensible filaments in low-Reynolds number flows. An extension of the traditional IBM to include a stochastic stress tensor is also proposed in order to model the thermal fluctuations in the fluid in smaller scales. The proposed numerical scheme is validated by comparing the response of a single actin filament immersed in a thermally fluctuating fluid to the theoretical values obtained using the WLC model.

The remainder of the article is organized as follows. The WLC model is reviewed in

The mechanical behavior of semiflexible filaments is usually described by the WLC model [_{b}

Using the Equipartition Theorem [_{B}_{p}

For the filament under consideration here, it is assumed that _{p}_{q}

Since the filament is inextensible, the total arc-length of the filament remains unchanged under the influence of the fluctuations. Thus, the arc length d_{3} is expressed as

By way of background, recall that the ensemble average <_{γ} f

In the context of the present problem, if the filament is in equilibrium at temperature _{q}_{i}_{i}_{p}

The probability density function

The fluid and the immersed semiflexible filament constitute a coupled mechanical system. The inextensible filament's motion is driven by the fluid's velocity field, while, at the same time, the filament exerts force on the fluid, thus affecting its motion. The equations of motion that describe the coupled system are derived and discussed in the remainder of this section.

Consider an inextensible massless filament of length _{x}_{y}_{h}

Assuming quasi-static loading conditions, the equilibrium equations are written as
_{1} over the entire filament leads to

Following Moreau et al. [_{||} and _{x}_{hi}_{2} over the entire filament, use integration by parts, and invoke _{1} and _{i,y0} is the moment of the external force acting on the _{0} = _{2} over the domain ((_{j}_{z}_{x}_{y}_{j}_{j}_{j − 1}_{i − 1,∥}_{i ,∥}_{0} = _{0} .

In the low-Reynolds number regime, the hydrodynamic force experienced by the filament immersed in fluid with velocity field _{h}

The position _{i}_{k,∥}

There are now _{0}, _{0}, …, _{N − 1}

With slight abuse of notation, let _{i}

Note that, in general,

These coarse-grained elastohydrodynamics equations, in conjunction with _{0}, _{0}) = _{0}, and [

The matrix [_{i}_{ + 2,j} = _{i}_{ + 2,N + j} = _{i}_{ + 2,2N + j }= 0. Also, the column vector [

In addition, matrix [_{1}] and [_{2}] are _{N,2}

The system of equations in _{n + 1}^{n + 1}_{n}_{n + 1}

As one approaches smaller length scales, in the order of

To account for thermal fluctuations, the Cauchy stress tensor

The low-Reynolds number Navier-Stokes equations for an incompressible, Newtonian fluid with the additional stochastic stress tensor to account for the thermal fluctuations can be written, in the absence of body forces, as
^{−9}, in large part due to the cell's size, which is in the order of

Consider now an incompressible viscous fluid occupying a two-dimensional domain Ω and undergoing thermal fluctuations. The IBM formulation in its strong form can be understood as an enrichment of the two-dimensional Navier-Stokes equations accounting also for the forces generated by the deformation of the immersed body, with the linear momentum balance equations in _{2} for the fluid taking the form

In the discrete case, the force term _{hi}_{1} [_{e}_{e}

The implementation of the proposed numerical algorithm for simulating flexible inextensible filaments immersed in a fluid can be summarized as follows:

This section focuses on the mechanical response of a single massless filament immersed in a thermally fluctuating fluid and aims to validate the results of the computational model described in the previous section by comparison with the theoretical prediction based on the WLC model in

The biopolymers that comprise the cytoskeleton consist of aggregates of large globular proteins that are bound together rather weakly, as compared with most synthetic, covalently bonded polymers [_{t}

The inextensible filament is modeled by means of the CGM described in ^{−5} ^{−5} ^{−5} s is used. The algorithm described in

_{t}^{6} time steps, <Δ_{t}^{−13}. This compares very well to the theoretical value for the ensemble average obtained from ^{−13} for a relative error of approximately 4%. In view of this agreement, the comparison may serve as validation of the proposed numerical algorithm used to simulate the fluid-structure interaction of the immersed filaments under thermal fluctuations.

In this study, a modified and computationally efficient version of the Immersed Boundary Method, combined with the Coarse-Graining Method, was proposed for modeling inextensible semiflexible filaments in low-Reynolds number flows. Thermal fluctuations in the fluid were modeled by including a stochastic stress. The mechanical behavior of a massless, inextensible, and semiflexible filament immersed in a thermally fluctuating fluid was investigated using the suggested method. The resulting time-averaged contraction of the filament compares very favorably to the theoretical value for the ensemble average of the same quantity, as obtained from the Worm-like Chain model. On the basis of this analysis, the proposed hybrid algorithm appears to be both robust and accurate, and could offer a reliable means for investigating the combined effect of multiple (and possibly interacting) filaments in low-Reynolds number flows.