Coarse-grained model of the native cellulose $\hbox {I}\alpha$ and the transformation pathways to the $\hbox {I}\beta$ allomorph

Abstract

All-atom simulations are used to derive effective parameters for a coarse-grained description of the crystalline cellulose $\hbox {I}\alpha$. In this description, glucose monomers are represented by the C4 atoms and non-bonded interactions within the cellulose sheets and between the sheets by effective Lennard-Jones interactions. The parameters are determined by two methods: the Boltzmann inversion and through monitoring of the energies associated with changes of the coarse-grained degrees of freedom. We find that the stiffness-related parameters for cellulose $\hbox {I}\alpha$ are nearly the same as for $\hbox {I}\beta$ allomorph. However, the non-bonded terms are placed differently and are weaker leading to an overall lower energy, and free energy, of $\hbox {I}\beta$ compared to $\hbox {I}\alpha$. We apply the coarse-grained description to determine amorphous transition states for the room-temperature conversion process between the $\hbox {I}\alpha$ and $\hbox {I}\beta$ allomorphs and to characterize the interface between the crystalline forms of the allomorphs.

Introduction

Cellulose is one of the most abundant renewable biopolymers. It has been a subject of interest for a broad scientific community. Examples of the recent research are in the context of bioconversion of cellulosic wastes into fermentable sugars to make biofules (Bayer et al. 2007, 2010; Dashtban 2009; Peplow 2014) and in the design of high-performance materials that involve nanocellulose fibers (Lee et al. 2014; Håkansson et al. 2014).

Cellulose consists of unbranched homopolysaccharide chains of glucopyranose units, denoted as D-GLC. Two D-GLC units are linked via $\beta (1\rightarrow 4)$-glycosidic bonds which connect the C1 and C4 carbon atoms. The chains can combine into various allomorphic structures that may coexist. Two of them, $\hbox {I}\alpha$ and $\hbox {I}\beta$ are crystalline and their relative abundance ratio depends on the source of cellulose (Newman 1999; Kataoka and Kondo 1999). Cellulose $\hbox {I}\alpha$ is the main constituent of the cell walls of the green algae Cladophora (Mihranyan et al. 2007) and Valonia (Revol 1982), whereas the celluloses from the higher plants, such as cotton, wood and ramie are rich in the $\hbox {I}\beta$ allomorph. At one extreme, the cellulose produced by a marine animal, tunicate, is nearly 100 % $\hbox {I}\beta$ (Nishiyama et al. 2002). At another, the cellulose of the cell wall of the alga Glaucocystis can be considered as almost 100 % $\hbox {I}\alpha$ (Imai et al. 1999).

The crystal structures of cellulose $\hbox {I}\alpha$ and $\hbox {I}\beta$ allomorphs have been characterized by means of synchrotron X-ray and neutron diffraction (Nishiyama et al. 2003, 2008). In both allophorms, cellulose chains are parallel and aligned along the same growth direction (coinciding with the crystallographic c-axis) as shown in Fig. 1. It has been found that cellulose $\hbox {I}\alpha$ and cellulose $\hbox {I}\beta$ differ in mutual packing of the chains. The $\hbox {I}\alpha$ allomorph has a triclinic unit cell of P1 symmetry with a single cellobiose unit as basis with layers displaced along the c-axis by +c/4. $\hbox {I}\beta$ has a monoclinic unit cell of the $P2_1$ symmetry with two cellobiose units forming a basis—they are termed origin (OR) and center (CE). $\hbox {I}\beta$ is thus made of two different sheets which are displaced by +c/4 and $-$ c/4 in an alternating fashion. The differences between the two crystalline allomorphs are captured by the interchain radial distribution function (denoted as RDF) shown in Fig. 2. Details of the RDF calculation will be presented in section IIIB. The RDF used here describes the probability of finding a C4 atom in an infinitesimal volume element at distance r from another C4 atom on a different chain. The RDF patterns are seen to deviate in the locations and heights of the corresponding peaks. The shift in the locations is slight for the first and fourth peaks, but is more visible in the case of the second and third peaks. The first peak for $\hbox {I}\beta$ is about 40 % taller than for $\hbox {I}\alpha$ and is located near r of 6 Å. The difference in the height of the first peak in the RDF between the two allomorphs is attributed to a non-uniform distance distribution of the four closest neighbours (see Fig. 2b). In the case of $\hbox {I}\beta$, these neighbours remain at similar equilibrium distances (see Fig. 2c), whereas in $\hbox {I}\alpha$ they fluctuate within a much larger range (see Fig. 2d).

The two different crystalline structures necessarily come with distinct placements of the hydrogen bonds (HB). There is a relatively strong presence of the O3–H$\cdots$O5 intrachain and O6–H$\cdots$O3 interchain HBs which are responsible for the stability of the layered structures. Experiments (Nishiyama et al. 2003) indicate the absence of the O–H$\cdots$O interactions between layers, but they suggest a larger presence of the C–H$\cdots$O intersheet HBs in the $\hbox {I}\beta$ allophorm compared to the $\hbox {I}\alpha$ one. Thus, it is expected that some C–H$\cdots$O hydrogen bonds and van der Waals forces contribute to the larger stability of $\hbox {I}\beta$ over $\hbox {I}\alpha$.

Different coarse grained models of cellulose have been developed to address important questions connected to the processing of plant biomass. Among them we can distinguish those related to (1) characterization of crystalline and amorphous cellulose phases (Molinero and Goddard 2004; Wohlert and Berglund 2011; Fan and Maranas 2014; Srinivas et al. 2014; Queyroy et al. 2004; López et al. 2015), (2) the transition from cellulose $\hbox {I}\beta$ to cellulose $\hbox {III}_{I}$ (Bellesia et al. 2012), and (3) understanding of the protein–polysaccharide interaction (Poma et al. 2015; Bu et al. 2010). However, none of these models describes the $\hbox {I}\alpha$ allomorph nor it considers the interconversion process into $\hbox {I}\beta$ allomorph.

Fig. 1

a shows the 36-chain microfibril model of the cellulose $\hbox {I}\alpha$ allomorph as viewed in the plane perpendicular to the c axis (i.e. the polymerization axis). Axes a and b define the triclinic unit cell formed by one chain. b Shows the layer packing in cellulose $\hbox {I}\alpha$. The layers are displaced along the $\textit{c}$-axis by $+\textit{c}/4$. Monomers shown in blue indicate the first D-GLC monomers in each layer. c, d are similar to a, d respectively but they illustrate the structure of $\hbox {I}\beta$. In this case, the unit cell is monoclinic. It is formed by two chains named origin (OR) and center (CE). A minimum number of chains for each allomorph is represented inside the unit cell (dashed black line). Layers in the $\hbox {I}\beta$ allomorph alternate between $+\textit{c}/4$ and $-\textit{c}/4$ displacements. (Color figure online)

Fig. 2

a shows RDF for the C4 atoms in the crystalline cellulose $\hbox {I}\alpha$ (solid line) and $\hbox {I}\beta$ (dashed line). The positions of the first four peaks along the r-axis are indicated by the vertical arrows next to the symbol “$d_{i}$” where the subscript i ranges between 1 and 4. It counts the peaks in the ascending order. The green arrows correspond to $\hbox {I}\beta$ and those in the black color to $\hbox {I}\alpha$. b shows four cellulose chains surrounding one central chain in the C4 representation. The eight C4 atoms which are closest to an arbitrary C4 atom (in blue) from the central chain are connected by sticks. The four blue sticks connect to atoms which contribute to first peak in RFD—located at $d_{1}$. The four red sticks connect to atoms contributing to the second peak at $d_{2}$. c, d Show the the distributions in the values of $d_1$ for the four closest neighbors in $\hbox {I}\beta$ and $\hbox {I}\alpha$ respectively. The thicker lines corresponds to the neighbor for which the distribution is closest to the first peak position in the RDF. The results were obtained through all-atom simulations at room temperature. (Color figure online)

Here, we extend our previous coarse-grained (CG) description of $\hbox {I}\beta$ (Poma et al. 2015) to the $\hbox {I}\alpha$ allomorph to bring out differences in the effective dynamical parameters between the two structures. The bonded interactions within the chains are found to be about the same, but the strengths of the effective non-bonded couplings differ. We employ the Boltzmann Inversion (BI) approach (Meyer et al. 2000; Jochum et al. 2012) and our own energy-based (EB) method (Poma et al. 2015). Our previous study (Poma et al. 2015) has been focused on cellulose $\hbox {I}\beta$ and two hexaoses. While the BI and EB methods agree in their account of the bonded parameters they disagree when dealing with non-bonded terms. However, the EB approach is deemed to be more reliable as it yields parameters which agree with experimental data on the strength of HBs in solids (Steiner 2002). We test our CG model and its parameters by making comparisons to all-atom simulations. In particular, we consider the RDFs and demonstrate the agreement between the two descriptions for both allomorphs.

We then turn our attention to the phase transformations between the $\hbox {I}\alpha$ and $\hbox {I}\beta$ structures. Conversion of $\hbox {I}\alpha$ into $\hbox {I}\beta$ is usually accomplished through hydrothermal heating at the temperature (T) between 533 and 553 K. The process lasts for more than half an hour (Wada et al. 2003). On heating, cellulose $\hbox {I}\alpha$ gets converted to an amorphous high temperature phase that turns into the $\hbox {I}\beta$ allomorph upon subsequent cooling. This kind of experiment is the main reason behind the common belief that the $\hbox {I}\beta$ phase has a lower free energy than the $\hbox {I}\alpha$ one (Debzi et al. 1991; Matthews et al. 2011). The conversion to $\hbox {I}\beta$ has great implications for industrial processes such as the dilute acid pretreatment of poplar and switchgrass. It has been observed (Foston and Ragauskas 2010) that the relative amount of the cellulose $\hbox {I}\alpha$ form decreases while the relative amount of $\hbox {I}\beta$ allomorph increases with the duration of the process. A similar result has been reported for pretreatment of loblolly pines (Sannigrahi et al. 2008). The $\hbox {I}\alpha$ allomorph is used industrially more often than $\hbox {I}\beta$ because of its higher reactivity towards acetylation (Sassi et al. 2000). In particular, it has been used for the production of cellulose derivatives such as bioplastic (Nawrath et al. 1995), artificial textile fibers (Eadie and Ghosh 2011), medical products (Lin and Dufresne 2014), etc. However, extraction of cellulose $\hbox {I}\alpha$ from its sources (wood, corn stalk, cotton, etc) carries serious environmental costs and new methods to provide it are needed. One of the model organisms considered in the context of the environmentally friendly $\hbox {I}{-}\alpha$ biosynthesis is Acetobacter xylinum (Lee et al. 2013). It should also be noted that the transformations between $\hbox {I}{-}\beta$ and $\hbox {I}{-}\alpha$ are expected to be facilitated by bending the microfiber during its formation (Jarvis 2000). Currently, no industrial process seems to have incorporated bending into the production of low cost cellulose $\hbox {I}\alpha$.

The timescale involved in the conversion process is beyond the capabilities of current all-atom simulations aimed at testing this expectation. New computational methods involving collective variables (Yu et al. 2014; Martoňák et al. 2005) may enable such modeling in the future. In the past, all-atom simulation studies were carried out to induce the conversion only in one unit cell (with periodic boundary conditions) (Hardy and Sarko 1996; Kroon-Batenburg et al. 1996) whereas our CG model can capture larger-scale structural reorganization inside the fibril. We thus use our CG model to study the conversion process not through a temperature cycling but through switching of the CG non-bonded terms between the two phases. We provide the energy characterization and confirm that the $\hbox {I}\beta$ phase is indeed more stable. We also determine the pathway of the conversion and the transition state. The conformation corresponding to the transition state can be a good starting point to study properties of the amorphous cellulose. Obtaining an amorphous state through heating turns out to be impractical computationally.

Methods

We have considered fibers made of 36 chains, 80 D-GLC monomers each. The chains are packaged into a nearly hexagonal shape as in ref. (Ding and Himmel 2006; Srinivas et al. 2014). When determining the CG parameters, we focus on a central 7-chain subset of this system.

All-atom molecular dynamics simulations

The molecular dynamics (MD) simulations were conducted with version 2.9 of the NAMD package (Phillips et al. 2005). The crystalline fibril of cellulose $\hbox {I}\alpha$ was parametrized by GLYCAM-06 force field (Kirschner et al. 2008; Tessier et al. 2008). A solvation box was used with 120 000 TIP3P water molecules Jorgensen et al. (1983) to allow for structural relaxation of the ideal crystalline structure prepared with the cellulose-builder toolkit (Gomes and Skaf 2012). Periodic boundary conditions were used and the electrostatic terms were counted by employing the Particle Mesh Ewald method (Darden et al. 1993) with a grid spacing of 1 Å in all directions. Numerical integration of Newton’s equations of motion involved the time step of 1 fs and the atomic coordinates were saved every 1 ps for analysis. The system equilibration was carried out by first minimizing the energy in 1000 steps and then by implementing a short 0.5 ns run in the NPT ensemble to achieve the atmospheric pressure of 1 bar. The production runs were carried out in the NVT ensemble at 300 K and they lasted for 20 ns. The temperature was controlled by the standard Langevin algorithm and the pressure by the Langevin piston pressure control algorithm. The MDenergy plugin from the VMD package (Humphrey et al. 1996) was used to compute the contributions of bonded and non-bonded energies.

Coarse-grained simulations

In our previous work on $\hbox {I}\beta$ and hexaoses (Poma et al. 2015), we have discussed two choices of representing the sugar units by effective atoms: either the effective atoms are placed at the center of mass of the D-GLC unit or at the location of the C4 atom. We have found the latter representation to be more stable numerically. This approach is conceptually more akin to representing amino acids by the $\alpha \hbox {-C}$ atoms in many CG models of proteins. Furthermore, the C4-based description was shown (Poma et al. 2015) to be able to distinguish between cellohexaose and mannohexaose which are stereoisomers whereas the other representation could not. Therefore, we use the C4 representation in our CG model for cellulose $\hbox {I}\alpha$, as illustrated in Fig. 3. Our CG simulations follow the scheme used for proteins (Sułkowska and Cieplak 2007; Sikora et al. 2009) and were carried out with an implicit solvent at 300 K which corresponds to $k_BT=0.59$ kcal/mol ($k_B$ is the Boltzmann constant). The Langevin equations of motion are used for thermostating and mimicking the presence of the solvent. They are solved by a fifth order predictor–corrector scheme. We simulate each cellulose fibril for $5000\tau$ steps, where $\tau$ is of order 1 ns.

Bonded interactions and their effective strengths will be derived in the next sections. The strength of the effective non-bonded interaction, $\epsilon ^{eff}$, depends on the type of the HB. Our method involves reading of the positions of the C4 atoms from an initial microfibril cellulose structure and then using this information to derive the length parameters $\sigma _{ij}^{eff}$ for the Lennard-Jones (12–6) potential between the effective atoms.

Fig. 3

An MD snapshot of the first four D-GLC monomers in a sheet of cellulose $\hbox {I}\alpha$. The CG description involves using the C4 atoms as representing the monomers (blue beads). The relevant CG degrees of freedom are shown within the dashed rectangle: (1) $r_{12}$ represents the CG distance between the first two D-GLC’s, (2) $\theta _{123}$ is the bond angle formed by three consecutive D-GLC monomers and (3) $\phi _{1234}$ is the CG dihedral between the four D-GLC monomers. The O and C atoms are in the red and gray colors respectively. (Color figure online)

Boltzmann inversion method

The BI method (Meyer et al. 2000; Jochum et al. 2012) allows for determination of parameters in a CG model by focusing on some degrees of freedom, q’s, such as the distance between the effective atoms or the bond angles formed by three sequentially consecutive effective atoms. The assumption is that, in the canonical ensemble corresponding to temperature T, independent degrees of freedom obey the Boltzmann distribution $P(q)=Z^{-1}e^{-U(q)/k_BT}$. Here, $Z=\int e^{-U(q)/k_BT}dq$ is the partition function. P(q) can be determined through the atomistic simulation of the reference system. Once this is done, one can derive the corresponding effective potential U(q), also known as the potential of the mean force, through the inversion $U(q)=-k_{B}T \ln P(q)$. Note that Z enters U(q) only as an additive constant.

Energy-based approach for calculation of effective bonded interactions

An alternative method (Poma et al. 2015) used here is to fit the mean atomistic energies to the functional dependence on q as postulated in the CG model. This approach does not assume that the variables q are truly independent. The first example is the effective bond potential, $V_b^\mathrm{\alpha \beta }$ which is defined between two effective atoms $\alpha$ and $\beta$ in a biopolymer. The atoms are separated by a time-dependent distance $r^{\alpha ,\beta }=|\textit{R}_{\alpha }-\textit{R}_{\beta }|$ which, generically, will be denoted as r. We assume that

$$\begin{aligned} V_b^\mathrm{\alpha \beta }({R}_{\alpha },{R}_{\beta }|k_{r},r_{0})=\frac{1}{2}k_{r}\left( r^{\alpha \beta }-r^{\alpha \beta }_{0}\right) ^{2}, \end{aligned}$$

(1)

where $k_r$ is the spring constant and $r^{\alpha \beta }_{0}$ is the equilibrium length of the bond. These two parameters can be determined by evolving the atomistic system and monitoring its total energies, E, that correspond to narrowly defined bins in the values of r. These energies are expected to be distributed in the Gaussian fashion. We plot the mean value $<E>$ of the E’s obtained within specific bins at a given r, as illustrated in Fig. 4a for a chain of cellulose $\hbox {I}\alpha$ and $\hbox {I}\beta$ (interacting with neighboring chains). We find that the dependence is indeed parabolic and we determine the corresponding parameters. The elastic $k_r$ parameters are obtained for the seven chains, which pass through the center of the fibril cross section (the central chain and its six closest neighbors), and for each chain we considered the 20-residue segment [31–50].

The CG effective bond angle potential involves three consecutive atoms denoted here as $\alpha , \beta$ and $\gamma$. It is represented as

$$\begin{aligned} V_{\theta }^\mathrm{\alpha \beta \gamma }({R}_{\alpha },{R}_{\beta },{R}_{\gamma }|k_{\theta },\theta _{0})=V_b^\mathrm{\alpha \beta }+V_b^\mathrm{\beta \gamma }+\frac{1}{2}k_{\theta }(\theta -\theta _{0})^2 \end{aligned}$$

(2)

where $\cos (\theta )=\frac{r^{\alpha \beta }\cdot r^{\beta \gamma }}{|r^{\alpha \beta }||r^{\beta \gamma }|}$ is the angle between the three D-GLC monomers (see in Fig. 3). The first two terms on the right hand side of Eq. (2) are the effective bond potentials for molecules ($\alpha$ and $\beta$) and ($\beta$ and $\gamma$). The last term in this equation is the effective bond angle potential which is typically represented by the harmonic potential. The determination of the force bending constant ($k_{\theta }$) and the equilibrium angle ($\theta _{0}$) is similar to the determination of $k_r$ and $r_0$ except that now the three body energies are monitored and the terms $V^{\alpha \beta }_b$ and $V^{\beta \gamma }_b$ are subtracted to get E. The results for $\hbox {I}\alpha$ and $\hbox {I}\beta$ are shown in Fig. 4b. In this calculation, we have considered 20 D-GLC units in seven adjacent chains, which generates a list of 18 angles per chain so the the error bar is determined based on 126 angles.

In a similar way, the effective torsion potential can be described by

$$\begin{aligned} V_{\phi }^\mathrm{\alpha \beta \gamma \delta }({R}_{\alpha },{R}_{\beta },{R}_{\gamma },{R}_{\delta }|\bar{a},\ldots )=V_b^\mathrm{\alpha \beta }+V_b^\mathrm{\beta \gamma }+V_b^\mathrm{\gamma \delta } +V_{\theta }^\mathrm{\alpha \beta \gamma } + V_{\theta }^\mathrm{\beta \gamma \delta }+f(\phi ) \end{aligned}$$

(3)

where $\phi$ represents the torsion angle between the $\alpha ,\beta ,\gamma$ and $\delta$ atoms. In order to get the needed E’s, we first subtract all of the two-and three-body potentials and then determine the distributions of E’s within bins corresponding to $\phi$. In our previous work (Poma et al. 2015) we determined the functional form of $f(\phi )$ for a cellulose chain which fluctuates in vacuum. In the absense of interchain and intersheet HBs, $f(\phi )$ is oscilatory. However, for a cellulose chain which is a part of the crystalline fibril follows the quadratic form:

$$\begin{aligned} f(\phi )=\frac{1}{2}k_{\phi }(\phi -\phi _{0})^2 \end{aligned}$$

(4)

This functional form agrees with other studies carried by Srinivas et al. (2014) and Fan and Maranas (2014) for cellulose $\hbox {I}\beta$. Our resuls are shown in Fig. 4c for cellulose $\hbox {I}\alpha$ and $\hbox {I}\beta$. We considered 20 D-GLC units for the seven chains and obtained a list of 17 dihedral angles per segment. The error bar are determined based on 119 dihedral angles.

The non-bonded interactions (the HB’s and ionic bridges) are represented by the Lennard-Jones potentials with the depth of the potential well $\epsilon$ and the length parameter $\sigma$. For small deviations away from the equilibrium this potential is equivalent to an effective harmonic term with the spring constant $k_{nb}$ such that $\epsilon ^{eff}={k_{nb}(\sigma ^\mathrm{eff})^{2}}{36^{-1}(2^{-2/3})}$ and $\sigma ^{eff}= 2^{-1/6} r_0$. The parameters are obtained in analogy to the procedure for the bond potential: one gets $k_{nb}$ and $r_0$ by first fitting to the harmonic potential near the minimum of the mean force, and then one infers about the $\epsilon ^{eff}$ from $k_{nb}$.

Fig. 4

Effective potentials computed from all-atom simulations by the EB method for cellulose $\hbox {I}\alpha$ (right side) and $\hbox {I}\beta$ (left side) at $\hbox {T}=300\hbox { K}$. The red curves correspond to the parameters listed within the parentheses. The inset shows the atomistic energy distributions corresponding to the data point surrounded by the square. a Describes the two-body bond potential. b Corresponds to the effective three-body interaction describing the bond angle potential. c Shows the four-body interaction which describes the dihedral term. (Color figure online)

Results and discussion

Coarse-grained description of cellulose $\hbox {I}\alpha$

The CG models of cellulose fibril that were proposed so far employed between one to four effective atoms for each D-GLC monomer. The level of the CG resolution depends on the chemical or physical property which is under consideration. With more than two effective atoms, there is enough resolution to describe conformational changes within individual D-GLC monomers (e.g. rotation of pyranose rings around glycosidic bonds). The more detailed description is also more suitable for capturing atomic rearrangements during the allomorph conversion (Bellesia et al. 2012; López et al. 2015). On the other hand, a one effective atom resolution is suitable for modeling large conformational changes at microsecond time scales. Such time scales are relevant for the transition from the amorphous to the crystalline phases in a cellulose fibril (Srinivas et al. 2014) and for the studies of bending of long fibrils (Fan and Maranas 2014). Our choice for a one-atom CG model follows our previous work (Poma et al. 2015) in which a polysaccharide-protein system was represented by effective atoms placed at the positions of the $\alpha$-C atoms in the protein and at the C4 atoms in the polysaccharide. Similar to the $\hbox {I}\beta$ case (Poma et al. 2015), each D-GLC monomer in a chain belonging to the $\hbox {I}\alpha$ allomorph interacts with the two neigbouring monomers, mainly via two intrachain HBs: (1) O3–H$\cdots$O5 and (2) O2–H$\cdots$O6 (Fig. 5a), which are observed most frequently during all-atom simulations (see in Table 1). The typical flat-ribbon conformation of a single chain of cellulose is commonly associated with the presence of the intrachain HBs which restrain the motion of two neigbouring D-GLC monomers along the $\textit{c}$-axis. The cellulose chains are organized into sheets that are connected by interchain O6–H$\cdots$O3 HB (Fig. 5a). The interactions between the sheets are weaker than the interchain HBs, primarily because they are coupled by the C–H$\cdots$O HBs (Fig. 5b). These couplings will be discussed later.

Table 1 Hydrogen bonds occupancy per D-GLC unit during the MD simulation

Fig. 5

The types of HBs between two chains that are present in cellulose $\hbox {I}\alpha$ or $\hbox {I}\beta$. a Shows two D-GLC chains belonging to a sheet. The O3–H$\cdots$O5 and O2–H$\cdots$O6 intrachain HBs are formed between two monomers within a chain and the O6–H$\cdots$O3 interchain HBs are responsible for keeping the chains together. b Shows common intersheet HBs of the kind C–H$\cdots$O; four of them are highlighted by the the green surrounding rectangles. c, d Show the effective representation of four chains in $\hbox {I}\alpha$ and $\hbox {I}\beta$ respectively. The distances $d_i$ discussed in the paper are indicated: $d_1$ (in blue) and $d_2$ (in red) are intersheet, $d_4$ is within the sheet (along the b direction in $\hbox {I}\beta$ and along the direction of the vector difference ${\mathbf {a}}-{\mathbf {b}}$ in $\hbox {I}\alpha$), and $d_3$ (in green) is between alternate sheets. (Color figure online)

Our results pertaining to the effective couplings derived by using the BI and EB methods are shown in Table 2. The values for $\hbox {I}\beta$ are taken from ref. (Poma et al. 2015) and those for $\hbox {I}\alpha$ are new. The top part of the table refers to the bonded interactions and the bottom part to the non-bonded ones. For a given method of derivation, the bonded parameters are seen to be nearly the same. Except for the stiffness in the dihedral terms, the BI and EB-based results are close in values.

Table 2 CG force field parameters for the allomorphs $\hbox {I}\alpha$ and $\hbox {I}\beta$ in the C4 representation

The non-bonded interactions arising within a single chain are effectively included in the value of the bonded parameter $k_r$, as they mainly restrain the axial elongation between two D-GLC monomers. Thus the interchain HBs are of a bigger interest when comparing $\hbox {I}\alpha$ to $\hbox {I}\beta$.

It has been noted earlier that there are two non-bonded energy scales associated with the two kinds of the interchain HBs (Heiner et al. 1995; Wertz et al. 2010; Wu et al. 2014): (i) those within the planar sheets, mostly due to the O6–H$\cdots$O3 HB (Fig. 5a), which is known to be the strongest and (ii) intersheet HBs between adjacent sheets (Fig. 5b), mostly due to the C–H$\cdots$O couplings. The former are stronger and arise more frequently. Table 1 lists the most frequent planar interchain and intersheet HBs in cellulose $\hbox {I}\alpha$ and $\hbox {I}\beta$ as observed during our all-atom simulations. The $\hbox {I}\beta$ occupancies are seen to be larger than the corresponding $\hbox {I}\alpha$ ones. This result correlates well with the contributions of non-bonded energy terms to the total energy of the system. Table 3 gives the non-bonded energies associated with two central sheets. The interchain terms are summed up over the sheets and the intersheet are a resultant of couplings arising between the two sheets. In both categories, the non-bonded energies are lower for $\hbox {I}\beta$ than i$\alpha$. This statement also applies to the electrostatic contributions which indicates that the HBs in $\hbox {I}\beta$ tend to last longer than in $\hbox {I}\alpha$.

Table 3 Total all-atom non-bonded interaction energy ($E_\mathrm{nb}$) divided into electrostatic ($E_\mathrm{elec}$) and van der Waals ($E_\mathrm{VdW}$) energy contributions for the interchain and intersheet interactions

The CG parameters for the non-bonded potentials are listed in the bottom part of Table 2 (they take into account both the electrostatic and van der Waals terms in the atomistic calculations). Judging by the depth of the effective LJ potential, the $\hbox {I}\alpha$ allomorph is characterized by weaker couplings than the $\hbox {I}\beta$ allomorph. This result is independent of the method used, but it is only the values derived by the EB that are consistent with the typical HB energy scales found in other solid systems (Heiner et al. 1995; Wertz et al. 2010).

One can simplify the data in the table further by taking the following parameter for $\hbox {I}\alpha$ (see Fig. 5c): 7.3 kcal/mol for the effective energy (depth of the potential well) between D-GLC chains within a sheet ($d_4$), 1.9 kcal/mol for HBs between parallel sheets ($d_1, d_2, d_5, d_6, d_7$, and $d_8$) and 2.5 kcal/mol between alternate parallel sheets ($d_3$). The similar numbers for $\hbox {I}\beta$ (see Fig. 5c) are: 7.4, 2.3, and 3 kcal/mol respectively. We observe that the strength of the effective couplings is not a monotonic function of the distance between the C4 atoms in the monomers.

Note that a coupling between chains separated by one cellulose layer involves the i and $i+1$ monomers (the distance of $d_3$) in cellulose $\hbox {I}\alpha$, whereas in cellulose $\hbox {I}\beta \ i$ interacts with i. For the parameters in these couplings, the BI method works much worse than EB, mainly because it does not take into account spacial correlations between the atoms, such as described by the RDF. There is a more sophisticated version of the BI method that take into account the correlations—this is the iterative Boltzmann inversion method (Meyer et al. 2000; Jochum et al. 2012). However, the disadvantage of the iterative approach is the resulting non-analytical form of the effective potentials as already found for cellulose $\hbox {I}\beta$ fibril (Srinivas et al. 2014). Basically, the BI method gives only the optimal solution in the limit of a highly diluted system and clearly some limitation of this technique must arise when dealing with crystalline systems. Thus, this method leads to a factor-of-6 overestimation of the energy parameters for $\hbox {I}\beta$ and a factor of 5 for $\hbox {I}\alpha$, when it is compared with typical HBs in solids (Steiner 2002; Vashchenko and Afonin 2014).

Tests of the coarse-grained model

We now consider the two allomorphs of the 36-chain systems and evolve them for 5000 $\tau$ using the simplified parameters listed in the previous section. We determine the distributions of the values of the bonded degrees of freedom and the RDFs. We compare them to those determined by the 20-ns evolution within the all-atom description and find that the CG approach works very well. Figure 6 indicates that the distributions of the values of the bonded degrees of freedom ($r, \theta$, and $\phi$) are essentially identical, regardless of whether they are determined during the CG (red lines) or all-atom (black lines) evolutions. The distributions are Gaussian and their means and dispersions are nearly the same for each of the three degrees of freedom. In addition, the differences between the two allomorphs are small.

Fig. 6

Comparison of the CG simulations (red lines) with the atomistic ones (black lines): distributions of: a bond lengths, b bending angles and c torsional angles. The left panels are for cellulose $\hbox {I}\alpha$ and right panels for $\hbox {I}\beta$. (Color figure online)

In order to characterize the type of packing in each allomorph we have determined the RDF. For a system of N interacting particles in a volume V, the RDF is defined (Frenkel and Smit 2002; Allen and Tildesley 1993; Tuckerman 2010) as

$$\begin{aligned} \mathrm {RDF}(r)=\frac{\langle N(r)\rangle }{\langle N_\mathrm{ideal}(r)\rangle } \end{aligned}$$

(5)

where r is the distance between a pair of particles. In this equation, the numerator, $\langle N(r)\rangle$, is the average number of particle pairs found between r and $r+dr$. The denominator is the average number of particles in the same shell, assuming a fully random distribution at density $\rho =N/V$. This last quantity routinely can be expressed as $N_\mathrm{ideal}=4\pi (N_\mathrm{pairs}/V)r^2dr$ where $N_\mathrm{pairs}=N(N-1)$ (for non-distinguishable particles). Integrating the dimensionless RDF over the radial dependency yields the number of particles: $\int _{0}^{\infty }RDF(r)\rho 4\pi r^{2}dr=N-1$. The interchain RDF is a minor variation of the previous RDF definition—the same-chain particles are not taken into account. In practice, the interchain RDF has been determined in the following way. We first compute the time averaged RDF for a given monomer in chain i and then average it over the monomers in this chain. This step is repeated for all remaining 36 chains and the average over all chains is taken to represent the RDF for the whole fibril. Since we focus on the interchain cellulose RDF, the volume involved in the definition corresponds not to the full simulational box but merely to the space taken by the fibril ($L_x=46.6\,{\AA }, L_y=43.8\,{\AA }, L_z=432.0\,{\AA }$ for $\hbox {I}\alpha$ and $L_x=50.6\,{\AA }, L_y=42.0\,{\AA }, L_z=414.0\,{\AA }$ for $\hbox {I}\beta$). This is done both in the all-atom and CG calculations. The all-atom results have been shown in Fig. 2 and are now repeated in Fig. 7 to make comparisons with the CG simulations. We have performed two kinds of simulations. In the first kind, the initial fiber structure was generated with the cellulose-builder toolkit (Gomes and Skaf 2012) and then energy minimized. In the second kind, as a starting state we have used the snapshot obtained after a 1 ns all-atom MD run. We observe that it is the second CG kind that agrees very well with the all-atom results. We interpret this as an evidence that important conformational changes in the model fibril take place as a result of the time evolution after the energy minimization.

Additionally, the insets in Fig. 7a, c show the RDF computed for the initial crystalline state. The line pattern RDF corresponds to the ideal crystal structure at zero temperature. Notice that the first peak for crystalline $\hbox {I}\alpha$ is represented by four single lines, whereas for $\hbox {I}\beta$ only 3 lines contrinute to the same peak at finite temperatures. This result shows that among the four closest neighbours which contribute to the first peak in crystal $\hbox {I}\beta$, two of them are separated by identical distances. The second peak is represented by two lines, the third and fourth by a single line for each of the allomorphs. At finite T’s the peaks broaden, shift in position, and start to overlap.

Fig. 7

a, c Show the RDFs obtained through the atomistic and CG simulations of cellulose $\hbox {I}\alpha$ and $\hbox {I}\beta$ respectively. The lines in black correspond to the all-atom MD simulations. The convention used is the same as in Fig. 2a. The colored lines correspond to the two types of the CG simulations that differ in the initial state of the fiber. The blue line corresponds to the initial state being all-atom equilibrated for 1 ns. The red line corresponds to the initial state obtained by constructing the fibril and then performing the energy minimization. The CG data was calculated based on 5000 $\tau$. The insets compare the results of all-atom MD at the room T with the spikes corresponding to the initial crystalline positions derived by using the cellulose-builder toolkit (Gomes and Skaf 2012). b, d Show the cross-section of the cellulose fibril along the c-axis for $\hbox {I}\alpha$ and $\hbox {I}\beta$ respectively. The four distances highligthed in the cross-section correspond to the same distances as shown in interchain RDF profile. Note that distance $d_3$ in $\hbox {I}\alpha$ is established between two D-GLC monomers from different cross-sectional planes as indicated by the bead in the red color. Their values are given in the top-right corner. (Color figure online)

These two allomorphs posses distinct unit cells and hence different crystallographic parameters. Table 4 provides the experimental lattice parameters for $\hbox {I}\alpha$ and $\hbox {I}\beta$ allomorphs. The cellulose chains in $\hbox {I}\beta$ interact through HBs in sheets parallel to the plane formed by axes b and c (Fig. 1c) and are stacked in the direction parallel to the vector b; the intersheet spacing between two adjacent cellulose sheets is defined by D = a/2. The experimental value for this parameter is 3.89 Å at room temperature. The organization of cellulose chains in $\hbox {I}\alpha$ is very similar to $\hbox {I}\beta$: the chains are parallel to the c-axis and sheets are grouped in parallel layers. The experimental value for intersheet separation, D, is 3.91 Å in $\hbox {I}\alpha$ and 3.89 Å in $\hbox {I}\beta$. Table 4 shows the lattice parameters computed with the CG model. Our CG description was able to retain the difference in D between the two allomorphs: it is 0.03 Å instead of 0.02 Å. In addition, the efficiency of our CG model relative to all-atom simulations can be measured in term of the cost required to reach a nanosecond time scale when determining the RDF. We find that the CG simulation is $\sim$1000 times more efficient since it takes half an hour compared to 500 h on the same processor with the all-atom simulations.

Table 4 Lattice parameters for pure $\hbox {I}\alpha$ (top) and $\hbox {I}\beta$ (middle) fibrils obtained from experiment, all-atom and CG simulation

Coexisting $\hbox {I}\alpha$ and $\hbox {I}\beta$ regions in a cellulose fibril

Despite new advances in the understanding of the cellulose biosynthesis in higher plants (Guerriero et al. 2010; Harris et al. 2010; Li et al. 2014), it is still unknown how cellulose chains assemble together into fibrils. Explaining the coexistence ratios of $\hbox {I}\alpha$ to $\hbox {I}\beta$ in various plants appears to be even harder (Atalla and VanderHart 1999; Fernandes et al. 2011). As an example of this situation, we now consider a fiber which is $\hbox {I}\alpha$ on one end and $\hbox {I}\beta$ on another—see Fig. 8a—with a transition region in between. The long direction is taken to be along the c-axis. Numerical all-atom simulations of such interfaces are often difficult due to problems in setting up a well relaxed initial state which would not lead to divergent atomic forces. Here, we explore taking a CG-based approach. Its advantage is that it allows for longer time scales necessary to reach the relaxed state and the potentials involved are soft.

We construct the initial configuration of the coexisting allomorphs via translation and alignment along the c-axis of two separate fibrils $\hbox {I}\alpha$ and $\hbox {I}\beta$. Each fibril is made of 36 D-GLC chains, 80-monomer long each. The first 40 monomers are initially set in the MD-relaxed $\hbox {I}\alpha$ structure and the last 40—in the $\hbox {I}\beta$ one. Note that this initial fibril structure is not realistic, since the distance between monomers i and $i+1$ in one chain at the interface is much larger than the corresponding equilibrium distance in another.

We then assign specific CG potential parameters for each allomorph in the two segments and evolve the system until reaching a stationary state. The interface acquires a transitional character which extends between i of 39 and 42, as shown in Fig. 8. The lower panel of this figure refers to the root mean square fluctuations (rmsf) in the stationary state. They were first calculated for each atom and then averaged over cross-sectional planes corresponding to a given i. It is seen that the fluctuations in the $\hbox {I}\beta$ segment are about 30 % smaller than in the $\hbox {I}\alpha$ segment. The difference in rmsf is an intrinsic behaviour of each allomorph and it is also observed in the all-atom simulation. Notice that in the atomic simulations, rmsf pertains to the C4 atoms. On the other hand, in the CG model the fluctuations characterize motion of effective atoms which are located at the C4 atoms (but could also be located at the centers of mass of the monomers). Thus the two quantities should not be equal, but both show the identical trend.

Additionally, we have carried out a structural characterization for the effective atoms from the four layers at the interface by computing the crystallographic parameters based on their positions. In this way, we first estimate the unit cell parameters (lengths and angles) for all effective particles in the transition region according to the triclinic ($\hbox {I}\alpha$) and monoclinic ($\hbox {I}\beta$) unit cell definitions. Then we average each parameter to compare with the reference (single-segment) values obtained from the CG simulation for each allomorph. The results are shown in the bottom part of Table 4. We observe that a monoclinic interface is not well-defined due to a large deviation in the monoclinic $\gamma$ angle, whereas a triclinic representation for this region is accurate for triclinic lattice constants but not suitable for the triclinic angles.

Fig. 8

a Snapshot of the interface between the $\hbox {I}\alpha$ and $\hbox {I}\beta$ segments of the fiber. The interface region is within the dashed vertical lines and it comprises monomers 39 through 42. b shows the corresponding transition in the rmsf as a function of the monomer number as calculated within the CG model (the upper line) and the all-atom simulation (the horizontal segments below). The average rmsf in CG simulation is 1.2 and 0.8 Å for the $\hbox {I}\alpha$ and $\hbox {I}\beta$ segments respectively

Conversion of $\hbox {I}\alpha$ to $\hbox {I}\beta$ at room temperature

The CG description allows for an easy determination of the total potential energy of the two allomorphs in the fibril form. We find that $\hbox {I}\beta$ is indeed lower in energy than $\hbox {I}\alpha$. The energy difference in the stationary state is ${\Delta } E =\langle E_{\alpha }\rangle -\langle E_{\beta }\rangle =136.8\hbox { kcal/mol}$.

We now use the CG description to explore the energy landscape corresponding to the transitions between the two allomorphs at room temperature. We induce the transition by first bringing an allomorph to its equilibrium and then by switching the values of non-bonded parameters $\varepsilon ^{eff}$ and $\sigma ^{eff}$ to those corresponding to the other allomorph, as listed in the bottom part of Table 2. As a result, the system overcomes high energy barriers that are normally forbidden at room temperature. The energy changes involved are shown in Fig. 9a. The parameters shown in this figure were averaged over an ensemble of 100 induced trajectories for each system. Each trajectory was divided into the following stages: the first 500 $\tau$ prior the switching of CG parameters, then (2–6) $\tau$ corresponding to the transition regime whose exact duration depends on when all non-bonded interactions (NBI) are recovered, and after that—500 $\tau$. The error bar in the ground state energy of $\hbox {I}\beta$ for the transition $\hbox {I}\alpha \rightarrow \hbox {I}\beta$ is 2.72 kcal/mol per trajectory and 1.80 kcal/mol in the ensemble of trajectories, hereas for $\hbox {I}\alpha$ (in the reverse process $\hbox {I}\beta \rightarrow \hbox {I}\alpha$) the corresponding error bars are 3.92 kcal/mol per trajectory and 2.40 kcal/mol in the ensemble. The single-trajectory error bars are comparable to those of the ensamble-averaged data which reflects existence of correlations induced by the adoption of the same crystalline initial state.

Notice that the NBI in equilibrium is equal to 12861 for both allomorphs. To determine the transition state (TS) the change in the NBI in the parameter-switching trajectories. In this way, the TS state is uniquely defined by the minimum value of NBI on the pathway. In practice, a trajectory starts by first losing some of its initial NBIs. Note that the system has to go through high energy conformations in order to reach the final destination. We have verified that the simulation reached the final state compatible with the CG parameters by computing the interchain RDF for the final structure (data not shown). The TS for the $\hbox {I}\beta \rightarrow \hbox {I}\alpha$ transition has a higher energy and a smaller number of NBIs than for the reverse transition. Panels (b) and (c) in Fig. 9 show examples of snapshots of the TS for the two processes.

For the $\hbox {I}\alpha \rightarrow \hbox {I}\beta$ process the TS is characterized by a loss of about 5 % of the initial NBI. The reverse process $\hbox {I}\beta \rightarrow \hbox {I}\alpha$ leads to a TS characterized by twice the loss of about 10 % of the initial NBIs. Such losses affects packing of the chains and gives raise to amorphous regions across the fibril structure. Furthermore, by computing the time needed to go over the energy barrier, we find that a longer escaping time is needed to implement the process $\hbox {I}\beta \rightarrow \hbox {I}\alpha$ ($t_{\beta \rightarrow \alpha }\approx 2t_{\alpha \rightarrow \beta }$). This is consistent with $\hbox {I}\beta$ having a lower free energy than $\hbox {I}\alpha$.

We now assess the difference in the free energy between the CG representations of the two allomorphs by using the thermodynamic integration (TI) approach (Frenkel and Smit 2002). In TI, a hybrid potential function ($U_\mathrm{hyb}$) is defined as follows:

$$\begin{aligned} U_\mathrm{hyb}(\lambda )=\lambda U_{\alpha }+\left( 1-\lambda \right) U_{\beta }, \end{aligned}$$

(6)

where $\lambda$ is a coupling parameter. This parameter indicates the level of change that is taking place on switching between $\hbox {I}\alpha$ and $\hbox {I}\beta$. The potential $U_{\alpha }$ and $U_{\beta }$ correspond here to the CG energies of the two allomorphs respectively. The interactions are switched when $\lambda$ is continuously decreased. Simulations conducted at different values of $\lambda$ allow to plot a $\frac{\partial U(\lambda )}{\partial \lambda }$ curve, from which ${\Delta } F= F_{\alpha }-F_{\beta }$ is derived as below:

$$\begin{aligned} F_{\alpha }-F_\beta =\int _{0}^{1}\left\langle {\frac{\partial U_\mathrm{hyb}(\lambda )}{\partial \lambda } }\right\rangle _{\lambda } d\lambda , \end{aligned}$$

(7)

where $U_\mathrm{hyb}$ is the hybrid interaction and $\langle \cdot \rangle$ denotes the canonical (NVT) ensemble average. We have carried out CG simulation for 20 values of $\lambda$ ($0,0.05,0.1,\ldots ,1$) from which $U_\mathrm{hyb}$ was determined, then its derivative with respects to $\lambda$ was obtained (see in Fig. 10) and finally the free energy difference was computed via the numerical integration. The free energy diference, $F_{\alpha }-F_{\beta }$ is found to be about $\sim$1.3 kcal/mol. This value confirms that $\hbox {I}\beta$ has not only a lower potential energy than $\hbox {I}\alpha$ but also a lower free energy. Solution calorimetry measurements (Goldberg et al. 2015) for the conversion processes of different cellulose allomorphs from metastable amorphous samples into the $\hbox {I}\beta$, II and III allomorphs indicate that these processes are exothermic. Our findings support this observation by detecting an amorphous phase during interconversion between $\hbox {I}\alpha$ and $\hbox {I}\beta$ allomorphs. In addition, the measured free energy differences: $F_\mathrm{{I}\beta }\hbox {-}F_\mathrm{II}=1.10\pm 1.60$ kcal/mol and $F_\mathrm{{I}\beta }{-}F_\mathrm{III}=1.40\pm 1.90\hbox { kcal/mol}$ are of the same order as our estimates for the $\hbox {I}\alpha \rightarrow \hbox {I}\beta$ transition. This shows the adequacy of our CG parameters.

Fig. 9

a The energy landscape for the convertion process between $\hbox {I}\alpha$ and $\hbox {I}\beta$ as derived through the CG simulations. The energies are given in units of kcal/mol and conversion times in unit of $\tau$ (of order 1 ns). The numbers of non-bonded interactions (NBI) are indicated. The pure state energies were averaged in 100 trajectories for 500 $\tau$ before switching the CG parameters to another phase. The error bars for the average energies, $\langle E_{\alpha }\rangle$ and $\langle E_{\beta }\rangle$ are 2.40 kcal/mol and 1.80 kcal/mol respectively. The error bars in the transition state energies are 30.4 kcal/mol for the $\hbox {I}\alpha \rightarrow \hbox {I}\beta$ process and 45.6 for the reverse process. b and c show examples of snapshots of the transition state (TS) found for the conversion process $\hbox {I}\alpha \rightarrow \hbox {I}\beta$ and its reverse respectively

Fig. 10

The plot of $\langle \partial U_\mathrm{hyb}/\partial \lambda \rangle _{\lambda }$ as a function of $\lambda$ for a transition between $\hbox {I}\beta$ and $\hbox {I}\alpha$ using the thermodynamic integration method

Conclusions

We have presented a coarse-grained model which is able to describe two allomorphs of the native cellulose I. The effective parameters were derived by two methods (BI and EB). Regarding the effective bonded interaction we showed that both methods are consistent with each other. But for the non-bonded interaction, only the EB has shown to estimate the correct strength for interchain and intersheet HBs. Testing the CG force field in the study of pure cellulose phases $\hbox {I}\alpha$ and $\hbox {I}\beta$ has shown good agreement with crystallographic experimental data and all-atom simulation. An application for modeling of a nonideal cellulose fibril where cellulose $\hbox {I}\alpha$ and $\hbox {I}\beta$ coexist in equal proportion has shown that a smooth monotonic transition from $\hbox {I}\beta$ to $\hbox {I}\alpha$ takes place. Explorations (or transitions) performed between allomorphs within our effective description at room temperature have found amorphous states which are characterized by a lost of 5–10 % of the crystallinity. Our approach will allow us to study other ways in which $\hbox {I}\alpha$ and $\hbox {I}\beta$ may coexist. One scenario involves the $\hbox {I}\alpha$ coating the $\hbox {I}\beta$ cellulose fibril. Mechanical response of such a material to nanoindentation can be assessed within our approach at different ratios of $\hbox {I}\alpha$ to $\hbox {I}\beta$ at low computational cost.