1 Introduction

Along with experimental and theoretical investigations, computational modeling has become increasingly important in the fields of biomechanics and biophysics [1], thank to the following three facts. First, the speed of computation has been increased tremendously in the past decades due to the innovations of CPUs and parallel computing architecture, following the observations of Moore’s law. Supercomputers with special purposes for carrying out molecular dynamics (MD) simulations, such as Anton [2], have also been developed to enable the study of protein folding for milliseconds. Secondly, the detailed structures of proteins and cells have been extensively characterized using experimental techniques such as crystallography and super-resolution electron microscopes [3]. These characterizations have allowed us to build accurate computational models based on experimental measured topology and geometry. In addition, physical properties such as molecular stiffness and entropic elasticity have also been measured using state-of-the-art tools such as AFM and optical tweezers [4], so that they can be incorporated into the computational models. Thirdly, computational methods have also been advanced significantly along with the hardware development, such as the developments of multiscale modeling approaches [5, 6]. However, in many cases of biological studies, it is still challenging to rely on computational modeling to make discoveries and to guide design. The current main bottle neck is the lack of efficient computational methods to address the large separation of spatial and temporal scales between the properties of single molecules and those of bulk materials.

Continuum-based modeling such as finite element method (FEM) [7, 8], boundary element method (BEM) [9], finite difference method (FDM) [10], finite volume method (FVM) [11], and spectral element method (SEM) [12] have been successfully applied to study biomechanical and biophysical problems for organs, tissues, and cells [13–15]. These methods have been used extensively in different engineering fields and have become the mature and routine procedures, with commercial packages available for designing buildings, bridges, and automobiles [8]. However, due to the multiscale and heterogeneous nature of biological systems, it is often difficult to gain new insights with only continuum-based models. The problem is particularly severe for small systems such as the cell membrane and biopolymers in which thermal fluctuations strongly influence measurements and molecular properties.

On the other hand, MD has been employed to resolve many significant problems in small length scales, such as protein folding [5, 16]. However, the length and time scales accessible to MD are currently limited to nanometers in length and microseconds in time. Even with specially designed supercomputer [2], it can only reach millisecond to simulate proteins without complicated hierarchical structures. Although all-atom MD is able to provide new insights for problems with short characteristic lengths and times, many important biological processes for larger bio-molecules and cells that occur over seconds or even minutes are expected to be beyond all-atom MD for the next decade or longer.

To deal with various complex multiscale biological systems, a good starting point is to study some ‘model’ systems as examples. Red blood cells (RBCs) and deoxyribonucleic acids (DNAs) are good examples of biological systems with multiple characteristic lengths and times. The RBC is a model system for cell mechanics and biological membranes, while the DNA is a model system for biopolymers. Multiscale studies of RBCs span from the properties of membrane proteins to whole cell deformation. Several known protein mutations in the RBC membrane can result in altered mechanical properties of RBCs, leading to diseases such as hereditary spherocytosis [17]. For DNAs, the length scales span from the nucleobase (\(\sim \)1 Å) to the contour length (larger than 1 meter for the human genome DNA). The large variation in these characteristic lengths presents a severe challenge for bridging between continuum-based and atomistic-based models. Recently, several mesoscale modeling approaches emerged rapidly as ways to study these problems in which it is forbidden to use all-atom MD to simulate the entire system, while continuum-based modeling cannot reveal the mechanisms related to molecular details [18, 19]. In this review, we will briefly describe different mesoscale models for studying RBCs and DNAs.

2 Mesoscale modeling of red blood cell dynamics

Due to its simple membrane structure, red blood cells (RBCs, erythrocytes) have been studied as a model system for cell biomechanics [13, 14]. A RBC has no nucleus or other important organelle. It consists of hemoglobin solution and a biological composite membrane including a lipid bilayer and a cortex cytoskeleton [33]. There are about 33,000 Junctional Complexes (JCs) in the cytoskeleton, which is a basic protein unit, consisting of an actin protofilament connected by four to fix spectrin proteins. The cytoskeleton is attached to the lipid bilayer through the transmembrane proteins such as band 3 and glycophorin C. Furthermore, the transmembrane proteins can move freely in the lipid bilayer so that the cytoskeleton can slide against the lipid bilayer. The interaction between the lipid bilayer and the cytoskeleton is an important problem since it is related to physiological process such as RBC aging [34] and pathological mechanisms such as bilayer loss in hereditary spherocytosis [17], but this bilayer–cytoskeletal interaction remains unquantified due to its small length scale. It has been found that bilayer–cytoskeletal interactions affect RBC membrane fluctuations [19] and tank-treading motion in shear flow [35]. For example, different bilayer–cytoskeletal tangential friction coefficients in diseased and healthy RBCs can lead to different cytoskeleton density distributions and different bilayer–cytoskeletal interaction forces in shear flow [35]. In addition, only by accounting for bilayer–cytoskeletal vertical elastic interaction, the magnitude of membrane fluctuations can be predicted [19]. However, the RBC membrane is considered as a single continuum layer in most existing models in the literature [36–39]. Recently continuum-based and particle-based mesoscale models have been developed to consider different molecular details of the RBC membrane [20–25, 27–32, 35], as shown in Fig. 1. In the following, we will review three different mesoscale approaches as examples, including a three-level hierarchical multiscale continuum approach, a two-component dissipative particle dynamics approach and a two-component coarse-grained MD approach.

Fig. 1
figure 1

Continuum-based and particle-based (also called, atomistic-based) RBC models with different molecular details: One-component continuum whole cell model [20], two-component continuum whole cell model [21, 22], one-component spectrin-level whole cell model [23–25], two-component spectrin-level whole cell MODEL [26, 27], 3D junctional complex (JCs) model [28, 29], and two-component coarse-grained molecular-dynamics model [30–32]

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2.1 Two-component RBC model using three-level hierarchical multiscale continuum method

Peng et al. have developed a three-level multiscale model of RBCs and coupled it with a boundary element model to study the dynamic response of RBCs in tube and shear flows [21]. For the RBC membrane, we built a three-level multiscale model to simulate its viscoelastic behaviors at different length scales, including single protein scale, protein complex scale and cell scale. For the fluids surrounding the RBC, we used a boundary element approach to study the fluid dynamics based on the Stokes equation of viscous flow [22].

The dynamics of RBCs involves physics at different length scales. For example, in the whole cell level, RBCs may undergo tank-treading motions in shear flow in the micro meter scale [40, 41]. In the protein complex scale, JCs of RBCs may experience fluctuations due to mode switching [28, 42]. In the protein level, tension may induce spectrin unfolding in the nanometer scale [28, 43]. We developed a multiscale framework to investigate the physics in different length scales, including three models at three different length scales, and coupled them together using a hierarchical multiscale approach, as shown in Fig. 2.

Fig. 2
figure 2

Two-component RBC model using three-level hierarchical multiscale continuum method [21, 22]

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At Level I, Zhu and Asaro developed a model of spectrin using Monte-Carlo simulation [29]. Spectrin is the major RBC cytoskeleton protein that forms a triangular network. Rief et al. applied AFM to stretch a single spectrin and found that it can be modeled accurately using a nonlinear constitutive model called worm-like chain (WLC) [23, 24, 43]. Besides that, it was found that if the spectrin is stretched beyond its contour length, multiple protein domains may unfold to increase the contour length. In order to capture the full mechanical behavior of the spectrin, Zhu and Asaro took into account its domain unfolding feature by employing a Monte-Carlo method, as the probability of the unfolding is a function of the loading [29]. By simulating the unfolding of the spectrin, Zhu and Asaro got consistent force-displacement curves as in the AFM experiment [43]. In addition, the effect of stretching rate is also quantified. After these force-displacement curves are obtained, the information is passed as a database to the next level model (Level II).

At Level II, Zhu et al. developed a 3D JC model with molecular details using Brownian dynamics (BD), which is a basic protein complex unit in the RBC cytoskeleton [28]. In the JC, an actin protofilament is associated with four to six spectrins. Zhu et al. built an exact geometry model based on the state-of-the-art understanding of the molecular structure of a JC, including the binding sites between spectrins and actins [44]. They simulated thermal fluctuations of the RBC membrane using this model [28] and predicted the area and shear moduli of the cytoskeleton, as functions of membrane deformation [21, 28], which can be passed to the next cell level model (Level III).

At Level III, Peng et al. developed a whole cell model of RBCs using FEM and considered the normal and tangential interactions between the lipid bilayer and the cytoskeleton [21, 22]. The cytoskeleton may slide against the lipid bilayer [45, 46], since the cytoskeleton is connected to the lipid bilayer through transmembrane proteins such as band 3 and glycophorin C, which can move freely within the lipid bilayer. In order to describe the bilayer–cytoskeletal interaction accurately, we modeled the RBC membrane as two distinct layers. We employed an inner layer to present the cytoskeleton with finite area and shear stiffnesses, and negligible bending stiffness, and an outer layer to represent the lipid bilayer with zero shear stiffness, huge area stiffness, and significant bending stiffness. Vertical elastic interaction and tangential viscous friction between the lipid bilayer and the cytoskeleton are simulated using a contact algorithm based on the penalty method [47]. The bilayer–cytoskeletal friction coefficient is estimated based on the Stokes-Einstein relation and experimentally measured diffusivities of band 3 and glycophorin C in the lipid bilayer [48–50]. Reduced integration shell elements based on Mindlin theory were used to simulate both the outer and the inner layers [21].

Besides the elasticity, we also considered the viscosities of the lipid bilayer and the cytoskeleton, which play significant roles in dynamics processes such as tank treading. For example, it has been demonstrated that without considering the membrane viscosity, the predicted tank-treading frequency cannot match the experimental measurement [22, 51]. Different from the existing study [51], we consider the viscosities of the lipid bilayer and the cytoskeleton separately using a generalized Voigt–Kelvin stress–strain relation [14]. Furthermore, we modeled the sliding between the bilayer and the skeleton as a viscous friction force. The viscous fluids surrounding the cell are incorporated in the following boundary element model of Stokes flow.

We applied a BEM to simulate the Stokes flows inside and outside of the RBC and coupled it with FEM to study the fluid-structure interaction. We used a staggered algorithm with explicit time integration [22]. For the interface dynamics in the Stokes flow with zero Reynolds number, we applied the boundary integral equation [9] so that the velocity \({\mathbf {v}}^f\) is given as

(1)

where \(\Lambda \) is the viscosity ratio, \(\varGamma ^{fb}\) is the membrane surface and \(\bar{\mathbf {v}}^f\) is the prescribed undisturbed velocity field of the shear flow. The vector \(\Delta \mathbf {t}^f=\mathbf {t}^{f,1}-\mathbf {t}^{f,2}\) is the discontinuity in the interfacial surface traction, where \(\mathbf {t}^{f,1}\) is the traction in the outside surface \(\varGamma ^{fb,1}\) of the interface, and \(\mathbf {t}^{f,2}\) is the traction in the inside surface \(\varGamma ^{fb,2}\) of the interface. The surface traction is related to the nodal force through the principle of virtual work [52]. denotes the principal value integration.

\(\mathbf {G}\) is the Green’s function for velocity. Its components are

$$\begin{aligned} G_{ij}(\mathbf {x},\mathbf {x}_0)=\frac{\delta _{ij}}{\left| \mathbf {x} -\mathbf {x}_0\right| }+ \frac{(x_i-x_{0_i})(x_j-x_{0_j})}{|\mathbf {x}-\mathbf {x}_0|^3}, \end{aligned}$$
(2)

where \(\delta _{ij}\) is Kronecker’s delta. \(\mathbf {T}\) is the Green’s function for stress. It can be written as

$$\begin{aligned} T_{ijk}(\mathbf {x},\mathbf {x}_0)=-6\frac{(x_i-x_{0_i})(x_j-x_{0_j}) (x_k-x_{0_k})}{|\mathbf {x}-\mathbf {x}_0|^5}. \end{aligned}$$
(3)

Extensive validations have been conducted in each level to verify the models, such as optical tweezers stretching [21], micropipette aspiration [21], tank treading, and tumbling of RBCs in shear flow [22]. More importantly, the effect of stress-free cytoskeleton state on the tank-treading behaviors has been compared with the recent experiment in detail [26, 53, 54]. Besides validations, we predicted correlation between the occurrence of Sp unfolding and increase in the mechanical load upon individual skeleton–bilayer pinning points and related it to the vesiculation process [55]. The simulation results also show that during tank treading, the protein density variation is insignificant for healthy RBCs, but significant for cells with a smaller bilayer–cytoskeletal friction coefficient, which may be the case in hereditary spherocytosis [17]. We also predicted two different modes of motions for RBCs in shear flow [35] and studied the effect of stress-free state on the tank-treading motion [26, 53, 54]. We showed that the cell maintains its biconcave shape during tank-treading motions under low shear rate flows, by employing a spheroidal stress-free state in the cytoskeleton [26]. Furthermore, we numerically confirmed the hypothesis that, as the stress-free state approaches a sphere, the threshold shear rates corresponding to the establishment of tank treading decrease. By comparing with the experimental measurements [53], our study suggests that the stress-free state of RBCs is a spheroid that is close to a sphere, rather than the biconcave shape applied in existing models [26]. In addition, we also quantified the stability phase diagram of different motion modes in high shear rate flows and explored the effect of stress-free state on the phase diagram [54].

2.2 Two-component model of RBC membranes using dissipative particle dynamics

Besides the three-level hierarchical multiscale model of RBCs, Peng et al. also developed a DPD-based two-component model of RBC membranes by modeling the lipid bilayer and the cytoskeleton separately using two two-dimensional triangulated networks with \(N_v\) DPD particles [19]. We use DPD particles to model the fluids, which interact with the DPD network representing the lipid bilayer. We applied the bounce-back condition to enforce the non-slip condition between the bilayer and the fluids. In the following, we will describe DPD method for simulating the fluids first and then the details of the two-component RBC membrane model.

2.2.1 Dissipative particle dynamics

DPD is a mesoscopic numerical technique, which was originally developed to simulate fluids and polymers [56–58]. In the DPD simulations, fluids and polymers are represented by N DPD particles, which interact with each other via pairwise potentials and their motions follow Newton’s second law. Different from classical molecular dynamics, a DPD particle represents a cluster of atoms. Three kinds of pairwise interaction forces are applied between DPD particles i and j at positions \(r_{i}\) and \(r_{j}\), including conservative, dissipative, and random forces. They are written as

$$\begin{aligned} F_{ij}^{{C}}= & {} {a_{ij}}{\omega (r_{ij})}{\mathbf {e}_{ij}}, \end{aligned}$$
(4)
$$\begin{aligned} F_{ij}^{{D}}= & {} -{\gamma }{\omega ^{2}(r_{ij})}{(}{\mathbf {e}_{ij}}{\cdot } {\mathbf {v}_{ij}}{)}{\mathbf {e}_{ij}}, \end{aligned}$$
(5)
$$\begin{aligned} F_{ij}^{{R}}= & {} {\sigma }{\omega {(}r_{ij}{)}}{{\zeta }_{ij}}{{\Delta }{t}^{-1/2} {\mathbf {e}_{ij}}}, \end{aligned}$$
(6)

where \(\mathbf {r}_{ij}=\mathbf {r}_{i}-\mathbf {r}_{j},\,{r_{ij}}={\vert }{\mathbf {r}_{ij}}{\vert },\,{\mathbf {e}_{ij}}={\mathbf {r}_{ij}}/{r_{ij}}\), and \(\mathbf {v}_{ij}=\mathbf {v}_{i}-\mathbf {v}_{j}\). The strengths of conservative, dissipative, and random forces are specified by constants \({a_{ij}},\,{\gamma }\), and \({\sigma }\), respectively. \({\Delta }{t}\) is the time step size, and \({\zeta _{ij}}\) is a random number with zero mean and unit variance. \({\omega (r_{ij})}\) is a weight function given by

$$\begin{aligned} \omega \left( {r_{ij}}\right) =\left\{ {\begin{array}{ll} {1-r_{ij}/r_{c}} &{} \qquad r_{ij} < r_{c} \\ 0 &{} \qquad r_{ij} \ge r_{c} \end{array}} \right. \end{aligned}$$
(7)

where the cutoff radius is defined by \(r_c\). Due to the dissipation-fluctuation theorem, the coefficients of random and dissipative forces follow a relationship \(\sigma ^2=2\gamma k_{B}T\), where \(k_B\) is the Boltzmann constant and T is the temperature, so that the two forces act a thermostat. Furthermore, due to the soft core interaction in the conservative force, we are able to use much larger time steps than that in the MD simulations. In summary, DPD is efficient simulation technique for modeling fluids and polymers.

Fig. 3
figure 3

A Two-component DPD model of a RBC. B Local triangular networks of the two-component model. C Normal and tangential interactions between the lipid bilayer and the cytoskeleton. D Physical picture of the local bilayer–cytoskeletal interaction [19]

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2.2.2 Membrane viscoelasticity and bilayer–cytoskeletal interactions of the two-component RBC model

The RBC membrane is modeled by two different components in the two-component DPD model, i.e., the lipid bilayer and the cytoskeleton, as shown in Fig. 3. Each component is represented by a 2D triangular network with \(N_v\) DPD particles. The total elastic energy of the RBC membrane is written as

$$\begin{aligned} U = U_s + U_b +U_{a + v} + U_\mathrm{int}, \end{aligned}$$
(8)

where \(U_s\) is the cytoskeleton elastic energy, given by the WLC potential and a repulsive power potential as

$$\begin{aligned} U_s=\sum _{j\in 1 \ldots N_s}\left[ \frac{k_\text {B}T l_m (3 x_j^2-2 x_j^3)}{4p(1-x_j)}+\frac{k_p}{(n-1)l_j^{n-1}}\right] , \end{aligned}$$
(9)

where \(l_j\) and \(l_m\) are the spectrin natural length and contour length, \(N_s\) is the number of spectrins, while \(x_j=l_j/l_m\). p is the persistence length, \(k_\text {B}\) is the Boltzmann constant, T is the temperature, and \(k_p\) is the power term constant.

\(U_b\) is the elastic energy of the lipid bilayer, which is written as

$$\begin{aligned} U_b=\sum _{j\in 1 \ldots N_s}k_b \left[ 1-\cos (\theta _j-\theta _0)\right] , \end{aligned}$$
(10)

where \(k_b\) is the bending coefficient and \(k_b=2k_c/\sqrt{3}\), where \(k_c\) is the bending stiffness of the bilayer. \(\theta _0\) is the spontaneous angle and \(\theta _j\) is the instantaneous angle between two adjacent triangles as shown in Fig. 3B. Furthermore, \(U_{a + v}\) is the energy due to cell area and volume conservation, which is written as

$$\begin{aligned} U_{a+v}=\sum _{j\in 1 \dots N_t}\frac{k_l(A_j-A_0)^2}{2A_0} +\frac{k_v(V^\mathrm{tot}-V_0^\mathrm{tot})^2}{2V_0^\mathrm{tot}}, \end{aligned}$$
(11)

where \(N_t\) is triangle number, \(A_0\) is the initial triangle area and \(A_j\) is the current triangle area as shown in Fig. 3B. \(V_0^\mathrm{tot}\) is the initial volume of a RBC and \(V^\mathrm{tot}\) is the current volume. \(k_l\) and \(k_v\) are penalty constants.

In addition to the membrane elasticity, we also consider the viscosities of the lipid bilayer and the cytoskeleton by adding dissipative and random forces as

$$\begin{aligned} \mathbf {F}_{ij}^{D,k}= & {} -\gamma _k^T \mathbf {v}_{ij}-\gamma _k^C (\mathbf {v}_{ij}\cdot \mathbf {e}_{ij})\mathbf {e}_{ij}, \end{aligned}$$
(12)
$$\begin{aligned} \mathbf {F}_{ij}^{R,k} \Delta t= & {} \sqrt{2k_\text {B} T}\left( \sqrt{2 \gamma _k^T}d\overline{\mathbf {W}_{ij}^S} \right. \nonumber \\&\left. +\,\sqrt{3\gamma _k^C-\gamma _k^T}\frac{tr[d\mathbf {W}_{ij}]}{3}\mathbf {1} \right) \cdot \mathbf {e}_{ij}, \end{aligned}$$
(13)

where \(k=b,s\) stands for the lipid bilayer or the cytoskeleton, respectively. \(\gamma _k^T\) and \(\gamma _k^C\) are dissipative coefficients with a condition \(3\gamma _k^C>\gamma _k^T\); \(\mathbf {e}_{ij}\) and \(\mathbf {v}_{ij}\) are the relative position and velocity. \(d\overline{\mathbf {W}_{ij}^S}=d{\mathbf {W}_{ij}^S}-tr[d\mathbf {W}_{ij}^S]\mathbf {1}/3\), where \(d\mathbf {W}_{ij}\) is the Wiener increment. The Wiener increment \(d\mathbf {W}_{ij}(t)=\mathbf {W}_{ij}(t+\Delta t)-\mathbf {W}_{ij}(t)\) over a time step \(\Delta t\) is a random variable drawn from a normal distribution with zero mean and a time-step variance \({\mathcal {N}}[0,\Delta t]\). The membrane viscosities can be obtained as

$$\begin{aligned} \eta _k=\sqrt{3}\gamma _k^T+\frac{\sqrt{3}\gamma _k^C}{4}. \end{aligned}$$
(14)

Viscosity values are taken from the literature [59].

To describe the bilayer–cytoskeletal interaction, we add another term \(U_\mathrm{int}\) as

$$\begin{aligned} U_\mathrm{int} = \sum _{j\in 1 \ldots N_{bs}}{\frac{k_{bs}(d_j-d_{j0})^2}{2}}, \end{aligned}$$
(15)

where \(k_{bs}\) represents the elastic stiffness of the bonds between the bilayer and the cytoskeleton, and \(N_{bs}\) is the number of vertical bonds, including the bonds between the transmembrane proteins (band-3 and glycophorin C) and spectrins. We simulate the bilayer–cytoskeletal interaction by a tangential friction force and a normal viscoelastic force as shown in Fig. 3C, D. \(d_{j}\) is current bond length; \(d_{j0}\) is the initial bond length. Experiments show that \(d_{j0}\approx \) 30 \(\mathrm{nm}\) [60]. We employed a master–slave penalty contact algorithm to calculate the bilayer–cytoskeletal interaction force [47]. The elastic interaction force on a cytoskeletal vertex is given as

$$\begin{aligned} \mathbf {f}_j^{E} = k_{bs}(d_j-d_{j0}) \mathbf {n}_j, \end{aligned}$$
(16)

where \(\mathbf {n}_j\) is the normal unit vector. We also add a vertical damping force as

$$\begin{aligned} \mathbf {f}_j^{D}= -c_{bs}(\mathbf {v}_{j}\cdot \mathbf {n}_j)\mathbf {n}_j, \end{aligned}$$
(17)

where \(\mathbf {v}_{j}\) is the relative velocity and \(c_{bs}\) is the damping coefficient. The bilayer–cytoskeletal viscous friction force is given as

$$\begin{aligned} \mathbf {f}_j^{F}= -f_{bs}[\mathbf {v}_{j}-(\mathbf {v}_{j}\cdot \mathbf {n}_j)\mathbf {n}_j], \end{aligned}$$
(18)

where \(f_{bs}\) is the friction coefficient.

To keep the constant temperature, we add another random force [61] as

$$\begin{aligned} \mathbf {f}_j^{R} \Delta t= \sqrt{2k_\text {B} T}\left( \sqrt{2 f_{bs}} d\mathbf {W}_{ij}^A+\sqrt{3c_{bs}}\frac{tr[d\mathbf {W}_{ij}]}{3}\mathbf {1} \right) \cdot \mathbf {n}_j, \end{aligned}$$
(19)

where \(d\mathbf {W}_{ij}^A=(d\mathbf {W}_{ij}-d\mathbf {W}_{ji})/2\) is the anti-symmetric part of the Wiener increment.

To summarize, the total bilayer–cytoskeletal interaction force is given as

$$\begin{aligned} \mathbf {f}_j^\mathrm{int}= \mathbf {f}_j^{E}+\mathbf {f}_j^{D}+\mathbf {f}_j^{F}+\mathbf {f}_j^{R}. \end{aligned}$$
(20)

The two-component RBC DPD model has been validated against more than six different experiments by using the same set of input parameters, such as micropipette aspiration, membrane fluctuations, tank-treading motion in shear flow, bilayer tethering in channel flow, passage through microfluidic channel, and twisting torque cytometry. More importantly, based on this two-component DPD model, several controversies and issues in RBC mechanics have been resolved. The micropipette aspiration experiments of measuring both the area moduli of the cytoskeleton (\(k_s\)) and the lipid bilayer (\(k_l\)) were simulated accurately using the same set of input parameters [62–64], although these two area moduli differ by five orders of magnitude. It was found that due to the explicit incorporation of bilayer–cytoskeletal elastic interaction, the two-component model matches the thermal fluctuations experiments better than the one-component model [65]. It was also found that the dependence of the tank-treading frequency on the shear rate follows a linear relationship for a narrow channel but a nonlinear one for a wide channel, which resolves a controversy in the literature [66]. It has been demonstrated that the tank-treading motion is too fast for the bilayer–cytoskeletal slip to occur for healthy RBCs [45, 46]. The bilayer–cytoskeletal interaction strength was predicted by simulating the RBC tethering in the channel flow experiment [67]. In addition, using this two-component model, Li et al. also studied RBC deformability in a small microfluidic channel and investigated the membrane mechanical properties [68]. The twisting torque cytometry was simulated and it was found that the predicted rheological properties of the RBC membrane match well with experimental measurements. In these experiments, Li et al. also showed that bilayer–cytoskeletal interactions are important for cell membrane mechanical, rheological, and dynamical properties [68].

2.3 Two-component RBC model using coarse-grained molecular dynamics

In a shorter length scale, Li and Lykotrafitis developed a two-component coarse-grained MD model for RBC membranes by adopting the one-particle-thick solvent-free model [75] for the lipid bilayer and a WLC network [30] or detailed free-jointed chain model [31] for the cytoskeleton. Zhang et al. also applied a similar coarse-grained model to study the stiffening effects of nanoscale knobs on human RBCs infected with Plasmodium falciparum malaria parasite [69].

In the one-particle-thick solvent-free model of the lipid bilayer, the following interaction potential function was adopted to enforce the DPD particles in a monolayer to form a one-particle-thick layer without losing particles,

$$\begin{aligned} u(\mathbf {n}_{i},\mathbf {n}_{j},\mathbf {x}_{ij})= & {} u_R(r_{ij})+u_A(r_{ij}) \nonumber \\&+\,\alpha [a(\mathbf {n}_{i},\mathbf {n}_{j},\mathbf {x}_{ij})-1]u_A(r_{ij}) \end{aligned}$$
(21)

where \(r_{ij}\equiv |\mathbf {x}_{ij}|,\,\mathbf {x}_{ij}\) is the relative position vector between two particles, and \(\mathbf {n}_{i}\) is the unit directional vector of each particle. The attractive and repulsive potentials \(u_A(r_{ij})\) and \(u_R(r_{ij})\) are given by

$$\begin{aligned} \left\{ \begin{array}{l} u_R(r_{ij})=\epsilon \left( \frac{r_c-r_{ij}}{r_c-r_\mathrm{eq}}\right) ^8,\qquad \quad r_{ij}<r_c\\ u_A(r_{ij})=-2\epsilon \left( \frac{r_c-r_{ij}}{r_c-r_\mathrm{eq}}\right) ^4,\,\, \,\,\,r_{ij}<r_c\\ u_R(r_{ij})=u_A(r)=0,\,\,\quad \qquad r_{ij}\ge r_c \end{array}\right. \end{aligned}$$
(22)

where the coefficient \(\alpha =1.55\) is specified to maintain the correct bending stiffness, \(\epsilon =4.54k_BT\) is the depth of the energy well, \(r_c=2.6 d\) is the cut off length, \(r_\mathrm{eq}=2^{1/6} d\) is the equilibrium length, and d is the unit of length.

$$\begin{aligned} a= & {} (\mathbf {n}_{i}\times \mathbf {x}_{ij})\cdot (\mathbf {n}_{j}\times \mathbf {x}_{ij})\nonumber \\&+\sin \theta _0 (\mathbf {n}_{j}-\mathbf {n}_{i})\cdot \mathbf {x}_{ij}-\sin ^2 \theta _0. \end{aligned}$$
(23)

\(\theta _0\) is the spontaneous angle associated with the bilayer spontaneous curvature.

Fig. 4
figure 4

Two-component RBC model using coarse-grained MD [30, 31]. (A) The structure of RBC membranes. The spheres with different colors represent different components: junctional complex (red), lipid particle (blue), spectrin particle (gray) glycophorin particle (black), fixed band-3 (yellow), and mobile band-3 (green). (B) Undeformed membrane in topview. (C) Undeformed membrane in side view. (D) Deformed membrane in side view. (Copied from [31], Colour figure online)

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Table 1 Comparison of RBC models
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The translational and rotational motions are described by

$$\begin{aligned}&m_i \ddot{\mathbf {x}}_i=-\frac{\partial V_i}{\partial \mathbf {{x}}_i}, \end{aligned}$$
(24)
$$\begin{aligned}&I_i \ddot{\mathbf {n}}_i=-\frac{\partial V_i}{\partial \mathbf {{n}}_i}+\left( \frac{\partial V_i}{\partial \mathbf {{n}}_i}\cdot \mathbf {{n}}_i\right) \mathbf {{n}}_i- I_i (\dot{\mathbf {n}}_i \cdot \dot{\mathbf {n}}_i)\mathbf {n}_i, \end{aligned}$$
(25)

where \(V_i=\sum _{j=1}^N u(\mathbf {x}_{ij},\mathbf {n}_{i},\mathbf {n}_{j})\). \(I_i\) and \(m_i\) are the inertial moment and mass of a particle, and \(\mathbf {{x}}_i\) is the position vector of a particle. Li and Lykotrafitis applied similar interaction potentials between the lipid particle and band 3 particle.

To incorporate the cytoskeleton of spectrin, Li and Lykotrafitis first applied WLC model to simulate the spectrin between the band 3 and actin in the one-particle-thick model [30], each of which are represented by a CG particle. In a refined model, Li and Lykotrafitis employed 39 particles connected by unbreakable springs to represent the spectrin instead of a single WLC chain and also considered the interaction between the spectrins and the lipids by LJ potentials [31], as shown in Fig. 4.

The shear modulus predicted by their model is consistent with the experimental measurement [30]. The model has also been validated by comparing the results with experimentally measured viscosity and thermal fluctuations of the RBC membrane [31]. Li and Lykotrafitis found that the RBC membrane bending stiffness and the transmembrane protein diffusivity can be accurately captured by adjusting the interaction potential parameters [30]. They found that at higher strain rates the shear stress is mainly determined by the viscosity of the lipid bilayer, and at lower shear strain rates the shear stress is mainly due to the cytoskeleton. Using the extended model [31], they discovered that the pressure imposed by the cytoskeleton on the lipid bilayer is reduced due to the defects of the connections between the cytoskeleton and the lipid bilayer, which might occur in spherocytes. More importantly, by comparison, they showed that this pressure is even more significantly reduced if there are defects in the dimer-dimer association of a spectrin filament, which might happen in elliptocytes [17]. They also explored the attractive forces between the lipid bilayer and the spectrin filaments and showed that it leads to increased pressure from the cytoskeleton to the lipid bilayer.

2.4 Comparison of different RBC models and related experiments

The main features and applications of some existing mesoscale RBC models are summarized in Table 1. The main differences are the molecular details considered and the length scales these model can reach. The two-component CG model focused on the detailed bilayer–cytoskeletal interaction and transmembrane protein diffusion, but it is too expensive to employ it to study a whole cell with current methodology. The two-component DPD model studied the whole cell behavior and explore the bilayer–cytoskeletal in a simplified way. The three-level multiscale approach is developed to bridge three models in different length scale, but due to the information passing, some information is lost, such as thermal fluctuations in the whole cell level.

There are still plenty of room to improve these models. For example, only one-way coupling has been done for the three-level multiscale RBC model, and two-way coupling can be added to study how the deformation in the macroscale can influence the dynamics of the proteins in the microscale. In addition, future effort will be put on how to improve the efficiency of the two-component CGMD RBC model in order to apply it in the whole cell scale and coupled it with fluid motion. Moreover, smoothed dissipative particle dynamics (SDPD) and smoothed particles hydrodynamics (SPH) have been also applied to study RBCs models [76–78]. Compared to conventional DPD, SDPD adopts advantages from smoothed particles hydrodynamics (SPH) [79], such as thermal consistency and specification of viscosity.

Since the RBC is a model system for cell mechanics and biological membranes, RBC models can be applied in other systems as well. For example, recently we extended the two-component RBC DPD model to study the blood vessel walls by considering the isotropic matrix and the anisotropic collagen fibers separately [80]. Furthermore, we can also apply RBC models to study the lipid bilayer and the actin cortex network or the intermediate filament network in general eukaryotic cells, because some of them share similar structures as the spectrin network of RBCs, such as the auditory outer hair cells and the nucleus lamina [33, 81].

3 Mesoscale modeling of DNA dynamics

Mesoscale modeling of DNA has granted insights into DNA mechanics, conformation, and dynamics in free solution [82–94] and for DNA in nanoslits [95–111] and nanochannels [112–124]. In particular, there has been significant interest in the validation of polymer theory using DNA as a model polymer due to ease of direct observation of single DNA molecules and its highly monodisperse nature [104, 125–144], with potential applications in genome analysis and sequencing [145–150]. Here, we will focus on a challenging problem for mesoscale coarse-grained DNA models : DNA translocation through a nanopore with applications for DNA separation and sequencing.

Fig. 5
figure 5

The characteristic length scales of a DNA molecule range from angstrom to millimeters

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It is challenging to accurately model DNA molecules in solution due to the need to simultaneously resolve long-range interactions (i.e., hydrodynamic and electrostatic) and short-range interactions (i.e., local rigidity, hydrophobic, entropic). DNA properties of interest can span several orders of magnitudes in terms of length and time as shown in Fig. 5. The computational cost of calculating the continuum hydrodynamic field by solving the non-linear Navier–Stokes equation is typically of order O(\(N^{2.25}\)), where N is the number of mesoscale beads in the system, rendering computation of large systems prohibitively expensive. Recently, fully coupled mesoscale methods such as lattice Boltzmann (LB) [151, 152], fast-multipole integration [153, 154], and particle-based methods such as dissipative particle dynamics (DPD) [155, 156], and multi-particle collision dynamics (MPCD) [85, 157] have demonstrated O(\(N \log {(N)}\)) efficiency for capturing long-range interactions such as hydrodynamics interaction. These developments have spurred new interests in the investigation of DNA dynamics in microfluidic flow and other systems.

3.1 Coarse-grained DNA models

Mesoscale models of DNA dynamics in free solution often neglect DNA stiffness and sequence specificity, which are very important for understanding DNA properties in nano-confinement. There are two major bottlenecks for modeling DNA dynamics in nanoscale confinement: (1) the large number of degrees of freedom needed to capture short-range interactions and (2) the high cost to accurately calculate the long-range electrostatic and hydrodynamic forces. Novel multiscale methods that will allow efficient computation of long-range (hydrodynamic and electrostatic) and short-range (van der Waals) forces are needed to understand DNA dynamics in nanochannels and nanopores.

Efficient computation of DNA properties in different systems requires choosing a multiscale DNA model by a careful examination of the important physical interactions, as shown in Fig. 6. To model the properties of large DNA molecules in free solution, a flexible bead-spring polymer model coupled with a continuum fluid model may be used to accurately capture long-time DNA dynamics [91, 158, 159]. Each spring represents a large number of Kuhn segments. For small spring extension in the linear force-extension regime such as DNA equilibrium properties, the harmonic or finitely extensible non-linear elastic spring potential could accurately capture the extensional force [160, 161]. However, for large spring extension such as DNA under strong shear or extensional deformation, the WLC model is needed to accurately capture the DNA dynamics [82, 162].

In principle, a fully flexible chain could be used to model DNA properties of millimeter length scales. However, this inherently neglects intermolecular physics for length scales of order Kuhn length (\(\approx \)100 nm) and smaller due to DNA backbone stiffness. For DNA in nanochannels, DNA semi-flexibility due to the double-helix structure must be accounted for. A coarse-grained model that includes inter-segment bending potential to correspond to the DNA persistence length may capture DNA rigidity and the force-extension relation [106]. Recent studies have shown that the semi-flexible chain model captures DNA dynamics in nanoslits and nanochannels [101–103, 107–109, 119, 121–123]. However, the spring length of the semi-flexible chain is typically of length 2–10 nm, and the model is most useful for modeling shorter DNA molecules with contour length of order 10 \(\upmu \) m due to the computational cost. In order to study DNA dynamics in nanopores, the model must be able to capture nucleotide-pore and base–base interactions. Models such as the 3-site-per-nucleotide DNA model [86, 163, 164] have been shown to capture DNA hybridization kinetics and also DNA structural rigidity. It may be a promising start point to accurately model sequence-dependent DNA translocation through a nanopore.

Fig. 6
figure 6

Different scales of DNA model from a continuous string to atomistic

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Table 2 Comparison of DNA models
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3.2 Mesoscale modeling of DNA/polymer translocation through nanopores

Nanopore translocation has been extensively studied due to its promising application for rapid, label-free, and real-time DNA sequencing [165–171]. The length scale of interest covers that of a single nucleotide with size of \(s\sim 0.34\) nm to DNA persistence length (\(\sim \)50 nm) to that of the DNA contour length with N bases (N s can be more than several \(\mu \)m), as illustrated in Fig. 5. Similarly, the time scale ranges from that of the nucleotide diffusion time (\(\sim s^2 f_c/k_B T\), where \(f_c\) is the friction coefficient) to the translocation time (\({\sim }\)contour length/translocation velocity; translocation velocity \(\sim \) electrophoretic force/friction coefficient). Up to six orders of magnitudes of variation in characteristic length and time need to be captured, which is a severe challenge for current computational capabilities. In order to model DNA translocation through a nanopore, DNA interactions with the nanopore itself must be accurately included, and thus the smallest characteristic length scale must be smaller than the nanopore size. The conformation of DNA segments vary from rod-like (\(<\) \(50\) nm) to coil-like. Muthukumar first analyzed the translocation of a string-like polymer through a hole and found that the translocation time \({\sim } N^2\) without external force (thus diffusion dominated) and \({\sim } N\) for driven translocation [171]. More recent coarse-grained models that include hydrodynamic interactions and more detailed DNA structure have found slightly different power-law exponents [172–175]. Thus, different qualitative trends in the translocation dynamics are expected with the inclusion of more molecular details and hydrodynamic interactions.

Even so, all-atom modeling of even a short DNA fragments with only one persistence length (\(\approx \)50 nm) in solution would still be prohibitively expensive, requiring more than \(10^8\) atoms. Bhattacharya et al. applied all-atom MD to study DNA translocation through protein nanopore Mycobacteria smegmatic porin A (MspA). However, it takes days for using thousands of CPUs to simulate the translocation of ten nucleotides through a nanopore for 1 \(\mu \)s [176]. Li et al. also applied MD to study the translocation of a single-stranded DNA through nanopores on graphene membranes [177]. Another challenge due to the cost of performing one set of MD simulations is that it is difficult to collect large samples for statistical evaluation. On the other hand, coarse-grained models have been employed extensively to study polymer translocation through a pore [174, 175, 178–183]. However, it remains challenging to accurately incorporate nucleotide characteristics, hydrodynamic interactions, electrostatic interactions, and DNA-pore interactions together in a complete model. In order to overcome these challenges, two recently developed techniques, dissipative particle dynamics (DPD) and lattice Boltzmann (LB), are discussed here and the main characteristics are summarized in Table 2. The solvent modeling, DPD, LB, and continuum modeling are separately consistent with the chosen polymer model.

3.2.1 Modeling of DNA/polymer translocation using dissipative particle dynamics

In recent years, DPD has been applied to study polymer translocation through a nanopore. DPD was demonstrated to be efficient in modeling polymer translocation [183–185]. Kapahnke et al. first employed 3D DPD simulation to investigate the translocation of a polymer driven by a solvent quality contrast across different sides of a membrane [185]. Duong-Hong et al. applied DPD to study the combined DNA electrophoretic flow and electroosmotic flow (EOF) in nano-fluidic devices and considered the EOF effect by utilizing a slip velocity as the wall boundary [186]. They also compared the DPD simulation results with the experiments [186]. In order to consider explicitly the electrostatic interactions between particles and avoid the singularity caused by the Coulomb attraction between particles, hard-core interactions have often been employed to prevent [156, 187]. For example, in addition to the typical DPD interactions shown in Eqs. 4–6, Li et al. used hard-core Morse interaction and a screened Coulombic interaction in the following form [156]:

$$\begin{aligned} \mathbf {F}_{i}= & {} \sum _{i\ne j}\left( \mathbf {F}_{ij}^C+\mathbf {F}_{ij}^D+\mathbf {F}_{ij}^R+\mathbf {F}_{ij}^M +\mathbf {F}_{ij}^S +\mathbf {F}_{ij}^\mathrm{Coulomb}\right) +\mathbf {F}_{ij}^{ext},\nonumber \\ \end{aligned}$$
(26)

where \(\mathbf {F}_{ij}^C,\,\mathbf {F}_{ij}^D\), and \(\mathbf {F}_{ij}^R\) are the DPD forces shown in Eqs. 4–6. \(\mathbf {F}_{ij}^\mathrm{Coulomb}\) is the electrostatic interaction forcem, and \(\mathbf {F}_{ij}^M\) is a repulsive force for preventing overlapping of ions and counterions. \(\mathbf {F}_{ij}^\mathrm{ext}\) is the external force field.

The bond interaction between DNA segments is given as

$$\begin{aligned} \mathbf {F}_{ij}^{s}=k_\mathrm{bond} \left( 1-\frac{r_{ij}}{r_{0}}\right) \mathbf {n}_{ij}, \end{aligned}$$
(27)

where \(k_\mathrm{bond}\) is the strength of the harmonic bond and \(\mathbf {n}_{ij}\) is the unit direction vector of the bond. \(r_{ij}\) is the bond length, and \(r_{0}\) is the initial bond length.

Based on the DPD approach, He et al. [184] found that the translocation time also scales with the force field strength as \(\tau \sim \mathrm{E}^{-0.48\pm 0.01}\). For unforced translocation, it was found that the translocation time follows a scaling \(\tau \sim N^\beta \) with \(\beta \) approximate to 2.24 [185]. Guo et al. observed single-file and double-folded translocation processes by studying the conformational changes of the polymer during the translocation [188] and showed that the average translocation time steadily decreases with the increase of fluid flux. It was also found that the chain rigidity plays an important role on the dynamics of polymer chain translocation [189]. Feng et al. applied DPD with two different dissipative radii in three dimensions and compared with the results of Langevin dynamics (LD) method [187]. Their results showed that the scaling relationship between the translocation time and the polymer chain length is \(\tau \sim N^{0.94}\) by DPD method with small dissipative radius and \(\tau \sim N^{0.98}\) with large dissipative radius. It was also found that the translocation time predicted by LD is larger than that by DPD method. Li et al. showed that the average translocation time \(\tau \) scales with the polymer length N as \(\tau \sim N^\beta \) [190], where \(\beta \) depends on the solvent quality [156] and could be adjusted by varying the repulsive parameter \(a_{PS}\) between the polymer and the solvent in Eq. 4 as

$$\begin{aligned} a_{PS}=a_{ij}(1+\epsilon ), \end{aligned}$$
(28)

where \(\epsilon \) is a parameter that determines the solvent quality. Poor solvent quality can be modeled with \(\epsilon >0\), and good solvent quality is modeled with \(\epsilon <0\). A comparative study of DPD and modified LD was carried out for polymer translocation [191], and it determined that the inclusion of hydrodynamic interactions increases the translocation probability and decreases the translocation time. It was also found that the wall interfacial property plays an important role in the dynamics of polymer translocation. Uniform hydrostatic force and uniform electrostatic force have been applied to study the polymer translocation, and it was found that the scaling correlations \(\tau \sim \mathrm{E}^{-\xi }\) and \(\tau \sim N^\beta \) are valid only for coil-like chains [155]. The translocation process of star polymers through a nanochannel was investigated, and it was found that the translocation time shows a power-law dependence on the number of arms outside the channel and very weakly dependent on the number of arms in the channel [192].

Fig. 7
figure 7

Schematic representation of the DPD simulation model for a polymer chain and a narrow pore. In this figure, the neutral gray, light gray, and black color circles represent the polymer, solvent, and wall particles, respectively. In addition, the dark gray color circles represent the counterions. (Copied from [156])

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3.2.2 Simulation of electrostatic effect on DNA translocation using dissipative particle dynamics

There are different ways to consider the electrostatic effect in DNA translocation in the DPD simulations. One way is to considered the electroosmotic flow effect by utilizing a slip velocity as the wall boundary [186, 193]. But if the electrostatic interaction between particles are explicitly calculated, as shown in Fig. 7, there are two major challenges. The first challenging is that a soft core potential is usually used in the mesoscale modeling methods such as DPD, so that the ion and counterion may form a cluster pair due to the strong electrostatic interaction since there is no hard-core interaction to prevent overlap of ions and counterions [194, 195]. The second challenge is the computational cost due to the long-range feature of the electrostatic interaction.

A simple way to overcome these two challenges is to use a screened Coulombic interaction and another hard-core interaction to prevent overlap of the particles, as did in the work by Li et al. [156], but significant error may be caused by these assumptions. A screened Coulombic interaction is employed as

$$\begin{aligned} \mathbf {F}_{ij}^\mathrm{Coulumb}=\frac{q_i q_jl_B e^{-\kappa r_{ij}}}{r_{ij}} \left( \frac{1}{r_{ij}}+\kappa \right) \mathbf {n}_{ij}, \end{aligned}$$
(29)

where \(l_B=e_c^2/k_B T\epsilon \) is the Bjerrum length, \(\kappa \) is the inverse Debye length, and \(q_i\) and \(q_j\) are charges. In addition, a cut off length is defined so that this screened Coulombic interaction is only considered within this cut off distance. A hard-core Morse potential is applied to prevent ion clustering, which is given as

$$\begin{aligned} \mathbf {F}_{ij}^{M}=2D_e \alpha \left[ e^{2 \alpha (2r_s-r_{ij})}-e^{ \alpha (2r_s-r_{ij})}\right] \mathbf {n}_{ij}, \end{aligned}$$
(30)

where \(D_e\) is the interaction coefficient, \(\alpha \) characterizes the interaction range, and \(r_s\) is the equilibrium length. Hard-core potential similar to Morse potential can be also used. For example, Yong et al. recently applied LJ potential to prevent ion clustering in studying the translocation time dependence on channel length [196]. However, since the advantage of DPD is that much larger time step can be used than MD because of its soft core interaction, this advantage may be diminished if a hard-core potential is employed in DPD.

Instead of using a hard-core potential, Groot applied the ‘smeared charge’ model to prevent the clustering of oppositely charged DPD particles [195]. The charge is distributed as a linear symmetric cloud with decreasing density as

$$\begin{aligned} \rho (r)=\frac{3}{\pi r_e^3}\left( 1-\frac{r}{r_e}\right) , \end{aligned}$$
(31)

where \(r_e\) is the smearing radius and is usually set to \(r_e=1.6 r_c\) ,where \(r_c\) is the cut off radius in DPD. The electrostatic potential between these two charge beads is approximately given by

$$\begin{aligned} \begin{aligned}&\frac{4\pi r_eu(r)}{\varGamma r_c}\\&\quad =\left\{ \begin{array}{ll} \frac{52}{35}-\frac{4}{5}\left( \frac{rr_c}{r_e}\right) ^2 +\frac{2}{5}\left( \frac{rr_c}{r_e}\right) ^4-0.136\left( \frac{rr_c}{r_e} \right) ^{5.145},&{}\quad r<r_e/r_c\\ \frac{r_e}{rr_c}-3.21\left( 1-\frac{rr_c}{2r_e}\right) ^6,&{}\quad r_e/r_c<r<2r_e/r_c \nonumber \\ \frac{r_e}{rr_c},&{}\quad r>2r_e/r_c \end{array}\right. \end{aligned}\\ \end{aligned}$$
(32)

where \(\varGamma =e^2/(k_B T \epsilon _0 \epsilon _r r_c)\) is the coupling constant, e is the elementary charge, and \(\epsilon _0\) and \(\epsilon _r\) are absolute and relative electric permittivities. The charge is then applied on a grid, and the electrostatic field is solved locally by using a version of particle–particle particle-mesh (PPPM) algorithm by Becker et al. [197].Yang et al. applied a similar method to study the translocation by modeling explicit counterions and the solvent quality in a computationally efficient way and considered two different driving forces [155].

Gonzalez-Melchor et al. [198] applied a Slater-type charge density to consider the charge distribution around a particle as

$$\begin{aligned} \rho (r)=\frac{q}{\pi \lambda ^3}e^{-2r/\lambda }, \end{aligned}$$
(33)

where \(\lambda \) is the decay length of the charge.

The corresponding interaction potential between two charged particles is given as

$$\begin{aligned} \frac{4\pi r_eu(r)}{\varGamma r_c}=\frac{1}{r}\big [1-(1+\beta r)e^{-2\beta r}\big ], \end{aligned}$$
(34)

where \(\beta =r_c/\lambda \). To match Eq. 32 at \(r=0\), we will have \(\beta =52/(35r_e)\).

More importantly, they applied Ewald sum method to calculate the long-range interaction. Recently Wang et al. applied a non-uniform FFT-based Ewald summation method to calculate the long-range interaction based on Slater-type charge density in DPD simulations [199] and investigated the dendrimer-lipid interactions [200].

3.2.3 Modeling of DNA/polymer translocation using lattice Boltzmann method

Another popular methodology to model polymer translocation is to combine a coarse-grained model of particle dynamics [160] with Eulerian lattice Boltzmann (LB) fluid [201–205]. The particle and fluid can be coupled through the immersed boundary method (IB) or the bounce-back boundary condition [206–210]. The LB–IB method has been shown to capture the physical characteristics of polymer and particle deformation in shear flow, inter-particle hydrodynamic forces, and the near-wall hydrodynamic field  [116, 118, 202, 207, 211–213]. Direct comparisons with experimental measurements of DNA diffusivity and conformation in micro- and nanoslits have verified that LB accurately captures how DNA properties are affected by nano-confinement  [136, 214]. LB has also been shown to accurately predict DNA density distribution in microchannels  [215].

In order for LB to resolve solvent hydrodynamic interactions, the time between solvent particle collisions must be very short compared to the time for fluid hydrodynamics to propagate in the system, so the mean free path of the solvent is much smaller than the system size, i.e., the Knudsen number \(Kn=\lambda /L << 1\), where \(\lambda \) is the mean free path and L is the representative physical length scale of the system. Microfluidic systems are well suited for LB modeling as the dominant length scale is \(\upmu \) m, while the mean free path of water molecules is on the scale of nm. The validity of LB in the relevant parameter regimes is addressed in the context of a coarse-grained bead-spring model of DNA, which appears to satisfy that criterion. In addition, the Schmidt number (\(Sc=\nu /D_m\)), the ratio between the kinematic viscosity \(\nu =\rho /\eta \) (\(\rho \) is the fluid density and \(\eta \) is the fluid viscosity) and the coarse-grained bead diffusivity \(D_m\), must be large so that the fluid propagates hydrodynamic interactions between beads much faster than the beads can diffuse. For molecules undergoing microfluidic flow, the competition between the inertial transport and viscous transport of the fluid is characterized by the Reynolds number Re\(=\rho UH/\eta \), which is usually very small, where U is the flow velocity and H is the channel dimension.

LB is carried out on a three-dimensional cubic lattice with 19 discrete velocities (the D3Q19 model). The lattice spacing is \(\Delta x\), the kinematic viscosity is \(\nu \)=\(\eta /\rho \)=1/6\([\Delta x^2 / \Delta \tau ]\), and \(\Delta \tau \) is the LB time step. Details of the LB method can be found in recent reviews  [202, 208, 209, 216, 217]. The 19 discrete velocities \(\mathbf c _i\) are given by the following vectors: \((0,0,0),\,(\pm \, 1,0,0),\,(0,\,\pm \, 1,0),\,(0,0,\pm \, 1)\), \((\pm \, 1,\pm \, 1,0)\), \((\pm 1,0,\pm 1),\,(0,\pm 1,\pm 1)\). The magnitudes of the velocities are \(c_i=|\mathbf c _i|=0,1,\sqrt{2}\). The lattice simulation has grid size \(\Delta x\) and time step \(\Delta t\). The maximum velocity in the simulation is the speed of sound \(c_s= \Delta x/\Delta t\). The velocity distribution function \(n_i(\mathbf r , t)\) gives the fluid population with the velocity \(\mathbf c _i\) at position \(\mathbf r \) at time t. At equilibrium, the velocity distribution functions can be represented as a second-order expansion of the Maxwell–Boltzmann distribution, given by

$$\begin{aligned}&n_i^{eq}(\mathbf r , t) = \rho a^{c_i} (1+ (\mathbf c _i \cdot \mathbf u ) / c_s^2 \nonumber \\&\quad \quad \quad \quad +\,\mathbf u \mathbf u :(\mathbf c _i\mathbf c _i-c_s^2 \mathbf I )/(2c_s^4)), \end{aligned}$$
(35)
$$\begin{aligned}&\rho = \sum {n_i^{eq}}, \end{aligned}$$
(36)
$$\begin{aligned}&\rho \mathbf u = \sum \mathbf{c _\textit{i}\cdot n_\textit{i}^{\textit{eq}}}, \end{aligned}$$
(37)
$$\begin{aligned}&\varPi = \rho \left( \mathbf u \mathbf u +c_s^2\mathbf I \right) =\sum {n_i^{eq}\cdot \mathbf c _i\mathbf c _i}, \end{aligned}$$
(38)
$$\begin{aligned}&\sum {a^{c_i}c_{i\alpha }c_{i\beta }c_{i\gamma }c_{i\delta }} = c_s^4(\delta _{\alpha \beta }\delta _{\gamma \delta }+\delta _{\alpha \gamma } \delta _{\beta \delta }+\delta _{\alpha \delta }\delta _{\beta \gamma }). \end{aligned}$$
(39)

The indices \(\alpha , \beta , \gamma , \mathrm{and}\, \delta \) denote axis x, y, and z. \(\mathbf u \) is the local velocity and the coefficients \(a^{c_i}\) are determined by satisfying the equilibrium conditions and the isotropy condition in Eq. 35. For the D3Q19 system, \(a^0=1/3,\,a^1=1/18,\,a^{\sqrt{2}}=1/36\). In terms of the fluid hydrodynamic properties, the velocity distribution functions are transformed to the hydrodynamic moments, \(\mathbf M _q=\mathbf m \cdot \mathbf n \), where \(\mathbf M _q\) is the qth moment of \(n=(n_0, n_1,\ldots , n_{18}\)), and \(\mathbf m \) is the transformation matrix. The 19 moments of the velocity distribution function are the density \(\rho \), the momentum density \(\mathbf j =\rho \mathbf u \), the momentum flux \(\varPi =\rho (\mathbf u \mathbf u +c_s^2\mathbf I )\), and the kinetic energy flux that conserve energy (‘ghost’ moments). At each time step, the fluid particles undergo local collision at the lattice site, and the velocity distribution functions evolve as

$$\begin{aligned} n_i(\mathbf r +\mathbf c _i\mathrm{d}t, t+\mathrm{d}t)=n_i(\mathbf r ,t) + L_{ij}[n_j(\mathbf r ,t)-n_j^{eq}(\mathbf r ,t)],\nonumber \\ \end{aligned}$$
(40)

where L is a collision operator for dissipation due to fluid particle collisions such that the fluid always relaxes toward the equilibrium distribution. Local dissipation of the fluid momentum is justified if the particle mean free path is much shorter than the lattice size, i.e., \(Kn<< 1\). For small Kn and Mach number, Eq. 35 has been shown to be equivalent to the Navier–Stokes equation.

Hydrodynamic interactions between the DNA segments can be captured with the exchange and propagation of frictional momentum between the fluid and the polymer [202, 213]. Each bead on the polymer experiences a friction force

$$\begin{aligned} \mathbf{F}_f = -\zeta \big [\mathbf{u}_b(\mathbf r _b) - \mathbf{u}_f(\mathbf r _b)\big ], \end{aligned}$$
(41)

where \(\mathbf{u}_b(\mathbf r _b)\) and \(\mathbf{u}_f(\mathbf r _b)\) are the bead and fluid velocity at the bead position \(\mathbf r _b\), respectively, and \(\zeta \) is the bead friction coefficient. The fluid velocity at position \(\mathbf{r}_b\) is determined by a trilinear interpolation of the fluid velocity on the neighboring lattice points nn that enclose \(\mathbf{r}_b,\,\mathbf{u}_f(\mathbf{r}_b) = \sum _{\mathbf{r} \in nn} w_\mathbf{r} \mathbf{u}(\mathbf{r})\), where the weights \(w_\mathbf{r}\) are the coefficients of the normalized linear Lagrange interpolation polynomial. The frictional momentum density \(\Delta \mathbf{j} = -\mathbf{F}_{f} \delta t / \Delta x^3\) is transferred to the fluid.

With the assumptions involved in LB and IB, systematic errors can arise due to fluid compressibility and the momentum transfer between the particle and the fluid. With the immersed boundary method, the no-slip boundary condition is imposed on the particle surface, and the friction momentum is calculated based on first-order Lagrange interpolation of the fluid velocity at the bead position from the lattice fluid velocity. Thus, the accuracy of momentum coupling is of O(\(\Delta x^2\)). Furthermore, the fluid velocity from LB is of O(\(\Delta x^2\)) accuracy.

Reboux et al. applied a coarse-grained model to solve the electrokinetic equations at the Poisson–Boltzmann level for the microions and coupled it to a LB equation for the solvent hydrodynamics to study DNA translocation [218]. Fyta et al. studied the problem of polymer fast translocation through a nanopore [151]. Melchionna et al. investigated the probability distributions of the translocation time of the DNA through the nanopore by adopting various initial configurations and lengths [219, 220]. Izmitli studied the electric-field driven translocation of 21–210 \(\upmu \)m DNA with and without hydrodynamic interactions and found that the effect of hydrodynamic interactions is local and the initial configuration of the molecules is much more important in determining the dynamics of DNA is for fast translocation processes [172]. It was also found that the coupling of the correlated molecular motion to hydrodynamics causes significant acceleration of the translocation process [152], and wide pores can host a larger number of multiple biopolymer segments, as compared to smaller pores [221]. In addition, Alapati et al. found that translocation velocity mainly depends upon the applied potential difference rather than upon the electric field inside the nanopores [222–224].

4 Challenges and future directions

Although both RBCs and DNAs have been studied extensively using various computational methods, some important questions still remain unanswered, especially due to the correlation between physical and chemical processes in various length scales. For example, the stress-free shape of RBCs is not completely understood [53], and more experiments are required to quantify the exact stress-free state of RBCs, which is closely related to the biconcave resting shape [26, 54], which is the first of eight mysteries about RBC proposed by Hoffman [228]. More importantly, how the membrane protein mutations lead to RBC diseases such as hereditary spherocytosis and elliptocytosis remain unclear. This problem has been explored using coarse-grained MD for a small piece of membrane [30, 31], but currently it is computationally prohibitive to extend this model to study a whole RBC. It is also a challenge to simulate the bilayer vesiculation process in RBC aging and RBC diseases using existing models, since it involves topological changes of the lipid bilayer. It can be done in a small scale [31], but not in the cell scale. Furthermore, chemical reaction processes such as binding of ATP to the cytoskeleton [229] and polymerization of abnormal hemoglobin in sickle cell disease [230–232] have not been accurately connected to the mechanical models yet. It is critical to connect biophysical models and chemical kinetics models for a better understanding of RBC physiology, such as RBC’s role in relaxing capillaries for improved blood perfusion.

For modeling of DNA translocation through nanopore, one of the critical issues is to identify the variation of electric current signature for different base pairs for the purpose of sequencing. Although mesoscale modeling approaches such as DPD and LB can be used to capture DNA dynamics on length scales of a few nanometers, several challenges remain. The most obvious of which is the high computational cost of modeling even small systems. Advances in algorithm, parallelization, or GPU-accelerated methods may be explored to reduce the computational time. Another challenge is in determining how to incorporate sufficient molecular details while maintain the advantages of coarse-grained modeling. Computational modeling research will continue to test the predictions of different levels of molecular coarse-graining to validate the model predictions. In addition, a common difficulty for both DPD and LB is to match the ratio between the fluid momentum diffusion and the coarse-grained bead diffusion (Schmidt number Sc) between the coarse-grained and the molecular systems. In the case of DPD, the Schmidt number is about 1000 times lower than the actual physical value of water. This may introduce computational artifacts in the simulation of a dynamic nonequilibrium process such as DNA translocation through a nanopore using DPD and LB. As with all computational results of real systems, results and predictions need to be thoroughly and rigorously tested against real observations. Despite these challenges, important new insights of dynamic mechanisms may be gained from the coarse-grained modeling of DNA and RBC for the appropriately chosen systems. Finally, although coarse-grained simulations is capable of predicting the translocation dynamics and providing guidance for the appropriate parameters for conductivity measurements, accurate calculations of the single nucleotide conductivity are still needed in order to differentiate between different bases. Future advances will require the combination of coarse-grained model of molecular configurations during translocation with first-principle calculations of nucleotide conductivity in order to provide model predictions of conductivity during translocation.

Even as the computational capacities keep increasing, mesoscale models will continue to provide a bridge between continuum and atomistic methods to fill the gap in our understanding of molecular dynamics and material properties. The successful examples of RBC and DNA have demonstrated the capabilities of mesoscale coarse-grained modeling as a tool of understanding cellular and biomolecular dynamics.