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A Priori Crystal StructurePrediction of NativeCelluloses
Remco J. Vietor1
Karim Mazeau2
Miles Lakin2
Serge Perez1,21 Ingenierie Moleculaire,
Institut National de laRecherche Agronomique,
Rue de la Geraudie`re,BP 71627,
44316 Nantes Cedex, France
2 Centre de Recherchessur les Macromolecules
Vegetales,*Centre National de la
Recherche Scientifique,BP53,
38041 Grenoble Cedex,France
Received 14 May 1999;accepted 29 March 2000
Abstract: The packing of b-1,4-glucopyranose chains has been
modeled to further elaborate themolecular structures of native
cellulose microfibrils. A chain pairing procedure was
implementedthat evaluates the optimal interchain distance and
energy for all possible settings of the two chains.Starting with a
rigid model of an isolated chain, its interaction with a second
chain was studied atvarious helix-axis translations and mutual
rotational orientations while keeping the chains at vander Waals
separation. For each setting, the sum of the van der Waals and
hydrogen-bonding energywas calculated. No energy minimization was
performed during the initial screening, but the energyand
interchain distances were mapped to a three-dimensional grid, with
evaluation of parallelsettings of the cellulose chains. The
emergence of several energy minima suggests that parallelchains of
cellulose can be paired in a variety of stable orientations. A
further analysis consideredall possible parallel arrangements
occurring between a cellulose chain pair and a further
cellulosechain. Among all the low-energy three-chain models, only a
few of them yield closely packedthree-dimensional arrangements.
From these, unit-cell dimensions as well as lattice symmetry
werederived; interestingly two of them correspond closely to the
observed allomorphs of crystallinenative cellulose. The most
favorable structural models were then optimized using a
minicrystalprocedure in conjunction with the MM3 force field. The
two best crystal lattice predictions werefor a triclinic (P1) and a
monoclinic (P21) arrangement with unit cell dimensions a 5 0.63, b5
0.69, c 5 1.036 nm, a 5 113.0, b 5 121.1, g 5 76.0, and a 5 0.87, b
5 0.75, c 5 1.036 nm,g 5 94.1, respectively. They correspond
closely to the respective lattice symmetry and unit-celldimensions
that have been reported for cellulose Ia and cellulose Ib
allomorphs. The suitability of
Correspondence to: Serge Perez; email:
[email protected] grant sponsor: INRA, CARENET-2, and
CNRS* Associated with University Joseph Fourier, Grenoble.
Biopolymers, Vol. 54, 342354 (2000) 2000 John Wiley & Sons,
Inc.
342
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the modeling protocol is endorsed by the agreement between the
predicted and experimentalunit-cell dimensions. The results provide
pertinent information toward the construction of macro-molecular
models of microfibrils. 2000 John Wiley & Sons, Inc. Biopoly
54: 342354, 2000
Keywords: b-1,4-glucopyranose chains; packing; molecular
structure; native cellulose microfi-brils; crystal structure
prediction
INTRODUCTION
Many recent advances in the theory and application ofmolecular
modeling to the structural elucidation ofcarbohydrate and
carbohydrate polymers have pro-duced a wide range of useful
results.14 In combina-tion with experimental methods, computer
modelinghas become an integral part of the strategy for reveal-ing
three-dimensional structures, both in solution andin the condensed
phase. Nevertheless, the realm ofcarbohydrate modeling has tended
to emphasize in-tramolecular rather than intermolecular aspects.
Whendealing with materials in condensed phases, the mod-eling
technique can be combined with informationderived from electron and
fiber diffraction to enablequantitative solution of the
three-dimensional crystal-line structure.5 In addition to
rationalizing why theobserved crystalline arrangement is the
preferredform, a further goal is the prediction of all
stablethree-dimensional organizations accessible to
thepolysaccharide in a given conformation. It would alsobe
desirable to extend this predictive methodology toless ordered
systems such as gels, where chainchaininteractions may occur to
promote the formation ofthe so-called junction zones.
These topics require the development of generalrules for
analyzing the stability of certain interhelixarrangements. Several
authors have proposed meth-ods for investigating the interhelix
structure and en-ergy through nonbonded forces.615 These
proceduresinvolve a minimization of the interhelix energy.
Incontrast, we have developed a method where thehelices are
positioned so as to allow contact but notinterpenetration of the
van der Waals surfaces of thetwo helices. After the helices are
placed at the posi-tion of van der Waals contact for a given
helixhelixrotation and translation, the energy is calculated.
Thisprocedure takes considerably less computer time thanmethods
involving energy minimization, and has beensuccessfully applied to
synthetic polymers,16 and tothe polysaccharide chitin and starch.17
In this latterexample, the structure predicted to be most
stablecorresponds to a duplex of parallel double helices, asfound
in both the crystalline A and B allomorphs.18,19From these results,
an explanation of the transitionfrom the A to B allomorph has been
proposed.17
Native cellulosic materials are organized into mi-crofibrils in
which crystalline domains coexist with
amorphous zones. Little is known about the ultra-structures of
the amorphous zones. Incidentally, thedetailed crystal structure of
native cellulose (celluloseI) is still a matter of debate despite
more than 70 yearsof research effort. X-ray fiber diffraction
experimentsinitially lead to models based on two-chain or
eight-chain unit cells depending upon the source of thesample,20,21
with the eight-chain unit cell invoked toaccount for weak signals
in the diffraction pattern.Later experiments using
cross-polarizationmagic an-gle spinning proton nmr indicated the
presence of twoallomorphs in samples of native cellulose,
designatedIa and Ib.2224 Subsequent results from electron
dif-fraction25 verified the presence of these two allo-morphs and
provided data on crystal symmetry andunit cell size for each
allomorphs. The Ia allomorphcrystallizes in the triclinic P1 space
group and con-tains one cellobiose unit per unit cell with a
parallelarrangement of the chains as would be expected. TheIb form
crystallizes in the monoclinic space groupP21 with two cellobiose
moieties per unit cell. Sincethe two allomorphs are found within
the same micro-fibril, parallel packing for Ib is the inescapable
con-clusion. The chain repeat length is found to be invari-ant at
1.036 nm.
Building realistic macromolecular models ofcellulosic
microfibrils starting solely with informa-tion derived from the
fiber repeat distance is still adifficult task. One needs to
predict both crystallineallomorphic phases of cellulose together
with lessordered regions, which could occur in the amor-phous
phase. This requires an exhaustive explora-tion of the low-energy
three-dimensional arrange-ments of cellulose chains. The present
workassesses the feasibility of the methodology of gen-eration. A
very important aspect of the work is theadequation of the proposed
models. Therefore,prior to the generation of dense packed chains
oneneeds a proper validation of the method used; thiscan be
achieved by a careful prediction of the twoallomorphs of cellulose
I. In addition to refining themethodology for predicting solid
state polymor-phism, the study should yield the crystal structureof
the two cellulose I allomorphs, and provide someinsight into the
nature of each form and into tran-sitions between them.
Crystal Structure Prediction 343
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COMPUTATIONAL METHOD
Nomenclature
The atom coordinates for the glucose residue used in thisstudy
were taken from the MONOBANK database ofmonosaccharide
structures.26 The atom numbering and theangle definitions used are
shown in Figure 1. The relativeorientation of two glucose residues
was determined by threeangles: the glycosidic bond angle t, defined
by the atomsC1OO4OC49, and the two torsion angles f(O5OC1OO1OC49)
and c (C1OO1OC49OC59). In ad-dition, the torsion angle v
(O5OC5OC6OO6) was used todescribe the position of the primary
hydroxyl group.
Computer Generation of Cellulose Chains
The PFOS program27 was used to generate cellobiose unitsand to
calculate the conformational energy and the helicalparameters n
(number of residues per turn) and h (advancealong the helicoidal
axis per residue) as a function of f andc for the primary hydroxyl
conformations gg (v 5 2608),gt (v 5 1808), and tg (v 5 608; see
Table I). Thecalculations used the 1.036 nm cellulose fiber repeat
dis-tance established by x-ray diffraction together with a gly-
cosidic bond angle t of 117.5. A typical potential energysurface
is represented in Figure 2.
Strict helical symmetry of a macromolecular chain re-quires that
equivalent chemical units occupy equivalentpositions about the
molecular axis. Such model chains wereprepared with glycosidic
torsion angles falling at minima onthe potential energy surface and
generating helices having n5 2 and h 5 0.518 nm. This type of chain
will be referredto as helical with t 5 117.5, f 5 290, and c 5
2153.
Cellulose chains exhibiting a repeat distance of 1.036 nmcan
also be constructed by regular alternation between two setsof
glycosidic torsion angles. Such chains, without internalsymmetry,
will be referred to as translational. They wereobtained by setting
one glycosidic linkage to one of the min-imum energy conformations
determined with PFOS; with thenext linkage manipulated until
translational symmetry with therequired period of 1.036 nm was
obtained between residues iand (i 1 2). Relative chain energies for
the conformationsobtained in this way were estimated by averaging
the energiesdetermined with PFOS for the individual sets (f, c).
The chainhaving the lowest energy was selected for further study.
Con-formational parameters for the generated chains are collectedin
Table I. Coordinates of both cellulose helices are availableupon
request form the authors.
Prediction of the Stable ChainChainArrangements
The relative orientation of two commensurate chains (chainA and
chain B) oriented either parallel or antiparallel, re-
FIGURE 1 Schematic representation of the cellobioseunit. The
atoms belonging to the reducing residue areprimed. The relative
arrangement of the two glucose resi-dues is determined by three
angles: the glycosidic bondangle t defined by the atoms C1OO4OC49,
and the twotorsion angles f (O5OC1OO1OC49) and c(C1OO1OC49OC59). In
addition, the torsion angle v(O5OC5OC6OO6) was used to describe the
orientation ofthe primary hydroxyl group, which normally adopts one
ofthe three staggered positions: gauchegauche (gg), gauchetrans
(gt), and transgauche (tg).
Table I Conformation Parameters of Studied Chains(for v 5 260,
60; 180)
Chain
t F cc
(nm)()
Helix 117.5 290 2153 1.036Translational 117.5 277 2141 1.036
117.5 2102 2164 1.036
FIGURE 2 Rigid residue potential energy surface of cel-lobiose.
Iso-energy contours are shown at 1 kcal/mol inter-vals relative to
the global minimum. Superimposed uponthese contours are those
calculated for the helical parame-ters n 5 2 and h 5 0.518 nm .
Where the cellulose chainadopts true 21 helical symmetry (f, c)
must assume thevalues (290, 2153). This requirement may be relaxed
ifthe glycosidic torsion angles alternate between (f, c)15 (277,
2141) and (f, c)2 5 (2102, 2164).
344 Vietor et al.
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quires a set of four interhelical parameters: mA, a rotation
ofthe chain A about the helical axis from 0 to 360, mB, arotation
of the chain B about its axis from 0 to 360, Dx,and Dz, which are
taken as positive, and represent posi-tional shifts normal and
parallel to the identity axis, respec-tively. Dz is bounded between
0 and t (fiber repeat). Thespatial description of these parameters
is shown in Figure 3.
The minimum energy arrangement of the two polymerchains with
respect to a displacement, will tend to bring themolecules as close
as possible without interpenetration ofVan der Waals radii. In
reality, a small amount of repulsiveenergy resulting from
interpenetration of some atom pairscan be compensated by additional
attractions from the re-maining atoms pairs. However, nonbonded
contact dis-tances deviate by only 10% (0.020.03 nm) in
molecularsolids.28 The contacting procedure28 involves describing
thesurface of each chain by circumscribing a hard sphere of
appropriate van der Waals radius R around each of theconstituent
atoms. Then for a given orientation of the chains(as dictated by
rotation angles mA and mB and an incrementalong the chain axis Dz)
the relative translation Dx is foundthat will bring the surfaces
into a position where they are incontact, but without
interpenetration. In general, the finalposition of the two
initially penetrating polymer chains ischaracterized by the
following conditions:
1. For at least one atom pair (i, j) the ith atom of thechain A
is separated from the jth atom of the chain Bby the sum of the
associated van der Waals radii (Rij5 Ri 1 Rj). The atom pair i, j
that satisfies theseconditions is referred to as the determining
contact.
2. For all atom pairs in the two separate chains, there isno
pair at a distance closer than the sum of their vander Waals radii
(Dij . Ri 1 Rj).
3. Condition (2) is violated any time that an atom pair i,j is
involved in hydrogen bonding with no a priorioptimum value for the
interatomic distance. This lim-iting case is treated by identifying
all atom pairspotentially eligible for participation in interchain
hy-drogen bonding and omitting them from the proce-dure. Thus
hydrogen bonding will not violate princi-ple (1).
Using this contacting procedure for chainchain con-struction,
the search space is reduced to three geometricvariables. The
resulting interchain interaction energy (EAB)can then be calculated
to the required degree of approxima-tion.
For the simulation of cellulose, the interaction energy ofthe
two chains was considered to be the sum of all pairwiseatomatom
interactions and was calculated using a 612potential function,29,30
with an additional term to cover thestabilization arising from the
interchain hydrogen bonding.This was based on the distance between
the oxygen atomsthat can interact through hydrogen bonding
(0.250.30 nm)without recourse to the hydroxyl hydrogens.
The CHACHA program17 was used to map Dx and EABas a function of
the structural variables mA, mB, and Dz.The analysis was performed
by incrementing mA and mBover the whole angular range by increments
of a few de-grees and the relative translation (Dz) between the
twochains was studied over the length of the whole fiber
repeat,typically by increments of 0.010.05 nm (i.e., h/ 20).
Therotations mA and mB were set to be both independent andcoupled
(mA 5 mB 1 a). For each setting of the chain asa function of mA and
mB and Dz, the magnitude of theperpendicular offset Dx was derived
according to the con-tact procedure described above. The value of
the energyEAB was then computed. The mapping procedure was usedto
search for low-energy regions. In order to pinpoint theenergy
minima, regions containing a local minimum weresearched a second
time using intervals of 1 (m) and 0.005nm (for Dz). This procedure
provided a complete overviewof the symmetry (or lack of symmetry)
of the chainchaininteractions. The set of interhelical parameters
relate to thelattice symmetry that characterises the
three-dimensionalorganisation as follows:
FIGURE 3 Interhelical parameters used to define the geo-metric
orientation of the two parallel cellulose chains (Aand B): Chain
rotations mA and mB, interchain contactdistance Dx, and
longitudinal offset Dz.
Crystal Structure Prediction 345
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1. mA mB: Chain A and chain B are not related byany symmetry
operation. They are independent andboth form the asymmetric unit of
the cell.
2. mA 5 mB: Chain B is derived from chain A by asimple
translational operation.
3. mA 5 mB 1 1808, and Dz 5 0: The two chains areparallel and
related by a twofold operation. For atwofold screw axis operation,
the conditions will bemA 5 mB 1 1808 and Dz 5 t/ 2.
4. mA 5 2mB 1 1808: The two chains are antiparalleland related
by a twofold or by a twofold screw axissymmetry operation.
Several protocols were used for the grid search. In thesimplest
case, the rotations of the 2 chains were coupled (mA5 mB). For the
helical chains, the internal 21 screw sym-metry of the chains
reduced the range to be searched tobetween 0 and 180, and Dz
between 0.0 and 0.518 nm.This reduction of the search space was not
applicable to thetranslational chains.
In the study of the monoclinc arrangement, two casesneed to be
considered. In the first and most general case, thechain axes do
not coincide with the 21 screw axes of thelattice. This implies
that the chains have to be relatedthrough 21 symmetry, i.e., mA 5
mB 1 1808 and Dz5 0.518 nm (5c/ 2). In the alternative case, where
thechains do coincide with the screw axis, two groups of chainscan
be distinguished, with no a priori relation between thechains of
different groups; the relative orientations of twosuch chains are
independent. The chains within a group arerelated by translation in
the xy plane. For the helical chains,both of these situations are
possible, the first being a subsetof the second. For the
translational chains, only the firstsituation is possible.
Expansion to a Three-Dimensional Lattice
Given two independent orientations of B chains with respectto
the A chain, a three-dimensional lattice is described. Therequired
parameters, m1A, m1B, Dx1, Dz1, and m2 A, m2B,Dx2, Dz2, are
illustrated in Figure 4. Such arrangementswere determined by
combining a further single chain with achain pair from one of the
stable conformations obtained inthe previous step. The rotations of
the single chain and theduplex were always coupled since the
symmetry of thesystems under investigation does not allow more than
twoorientations of the cellulose chains. The vertical displace-ment
Dz was free for triclinic lattices, but fixed at 0.00 nmfor
monoclinic lattices.
The three-chain arrangements obtained in this way werescreened
to exclude those with a space-filling array of unitcell volume per
cellobiose outside the range of the experi-mental values, i.e.,
outside 0.3 and 0.4 nm3. The corre-sponding arrangements were
checked for stability using aSimplex optimization. Finally,
unit-cell parameters weredetermined for the refined arrangements
that gave the small-est unit cell volumes (i.e., the highest
densities). The de-tained procedure to calculate the unit-cell
parameters hasbeen described previously16 and the set of equations
to beused are given in the caption for Figure 4.
Up to five unit cells per series were retained. The
generalprocedure to predict the occurrence of all possible
three-dimensional arrangements, is summarized in Figure 5.
Simplex Calculations
The CHACHA chainchain algorithm allows three-chainarrays to be
considered stable even where two of the chainsdo not interact. This
situation would lead to channel-likevoids in the corresponding
crystal lattice and were thereforeexcluded.
In order to confirm their viability, Simplex optimizationswere
performed on the CHACHA-derived three-chain ar-rangements. The
interchain distances (Dx), the chain orien-tations (m), the
relative shifts (Dz), and the angle in the xy
FIGURE 4 Building up crystalline structures. The calcu-lations
for three chains gave the relative positions of thechains in stable
triplet interactions. For triclinic lattices, thesides a and b, the
unit cell angles a, b, and g, the projectionof g on the xy plane
(g*), and the lattice volume percellobiose unit can be calculated
from the parameters for thestable configurations using Eqs. (1)(7).
For the monocliniclattices, Eqs. (8), (9), and (10) were used
instead of (1), (2),and (6) respectively to calculate a, b, and
g.
a 5 Dx12 1 Dz1
2 (1)
b 5 Dx22 1 Dz22 (2)
tan a 5Dz2Dx2
(3)
tan b 5Dz1Dx1
(4)
g* 5 1808 2 m1, A 1 m2, A (5)cos g 5 cos a 3 cos b 2 sin a 3 sin
b 3 cos g* (6)
V 5 Dx1 3 Dx2 3 c sin g* (7)
a 5 Dx22 1 ~Dx1p2!2 2 4 3 Dx1 3 Dx2 3 cos g* (8)
b 5 Dx2 (9)
sin~1808 2 g! 52Dx1
asin g* (10)
346 Vietor et al.
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plane determined by the three-chain arrangement were var-ied
(eight parameters). In order to retain the required trans-lational
symmetry for the triclinic cases the rotation aroundthe helical
axis was held identical for the three chains. Thisgave 6 degrees of
freedom in total. The Simplex was gen-erated by multiplying the
initial value of each variable inturn by 1.05 and using the
obtained tupel as an additionalvertex for the simplex, giving 7
vertices in total. Simplexoptimisation was continued until the
minimum for the totalinterchain energy was reached. Both nonbonding
and hy-drogen-bonding interactions were taken into account forthese
energy calculations. Structures that showed large dis-placements or
rotations were rejected.
Minicrystal Generation and EnergyMinimization
The most favorable structural models (i.e., those having
thelowest unit cell volume) were optimized using a
minicrystalprocedure as described by French et al.31 Minicrystals
con-sisting of 7 cellotetraose units were generated using
thecalculated chain orientations and lattice parameters, in sucha
way that a central cellobiose unit was completely sur-rounded. The
resulting model was allowed to relax underthe MM3 force field.32,33
The total steric energy calculatedby MM3 includes intramolecular
terms (bond stretching andbending, forming and deforming pyranose
rings), as well asnonbonded forces that can apply to both intra-
and intermo-lecular interactions. The more stable arrangement will
havea lower total energy, regardless of whether the stabilitycomes
from a more stable isolated molecule or from a betterintermolecular
arrangement. Instead of energy contributionsestimated from explicit
atomic charges, the dipoledipoleenergy is evaluated. This is very
dependent on the dielectricconstant as is the hydrogen-bonding
term. A value of 4 wasused to mimic the effect of a crystalline
environment onisolated molecules. After relaxation, deformation of
the
central cellobiose unit was measured by calculating the rmsatom
displacement of all nonhydrogen atoms and of thefinal values for t,
f, and c. The orientations of the primaryhydroxyl groups after
relaxation were also determined. Thelattice energy of the central
disaccharide was estimated asthe sum of all intermolecular
interactions involving thisdisaccharide.
RESULTS
Chain Construction
Among the (f, c) combinations compatible with the1.036 nm fiber
repeat, only the one giving n 5 2 andh 5 0.518 nm falls within the
fully allowed zone ofthe potential energy surface shown in Figure
2. If anideal twofold helical structure is assumed for thecellulose
chain, there exists only one set of glycosidictorsion angles that
can generate the helical parame-ters. For such a helical chain,
minimal glycosidicbond energy is obtained with f 5 290 and c5 2153.
Further stabilization is possible throughinterresidue hydrogen
bonds between O5 and O39(oxygenoxygen distance, dOOO, approximately
0.25nm) and, depending on the orientation of the primaryhydroxyl
group, between O6 (orientation gt) and O39(dOOO 5 0.31 nm) or O2
and O69 (orientation tg;dOOO 5 0.28 nm). Consideration of the
potentialenergy surface suggests that slight deviations from
theideal helical structure could not only be accommo-dated but also
be energetically more stable. A chainnot conforming to helical
symmetry is characterisedby a set of alternating torsional
glycosidic angles (f,c)1 and (f, c)2, and is termed a translational
chain.Minimum energy for such a chain was found for thecombination
of (f, c)1 5 (277, 2141) and (f, c)25 (2102, 2164).
Packing of Helical Cellulose Chains
Two-chain Arrangements. Both coupled chain rota-tions and
independent rotations were considered and anumber of stable
arrangements were found. As a firststep, all possible arrangements
occurring between twoparallel cellulose chains, were examined. This
wasperformed by rotating mA and mB over the wholeangular range from
0 to 360 and the relative dis-placement of the two chains was
investigated over thewhole length of the fibre repeat and with the
differentorientations of the primary hydroxyl groups. For
eachsetting of the chains as a function of mA, mB, and Dz,the
magnitude of the offset perpendicular to the chainaxis Dx was
computed according to the contact pro-cedure. The value of the
energy corresponding to eachset of chain orientations was evaluated
as a function
FIGURE 5 Flow chart showing the general procedureused to
investigate all possible three-dimensional paralleland antiparallel
arrangements of cellulose chains.
Crystal Structure Prediction 347
-
of the set of four interhelical orientations. The searchof
energy minima was performed within the three-dimensional (mA, mB,
Dz) space. Seventeen energyminima were found within a difference
energy win-dow of 2 kcal/mol/cellobiose. The results indicate
thatthe significant energy minima occurred for values ofmA lying in
the vicinity of mB. This suggests thatappropriate packing of
neighboring helical chains canbe achieved with operations of simple
translation.
The following steps were conducted assuming mA
5 mB. This allowed for a straightforward two-dimen-sional study.
The contour maps calculated as a func-tion of the translation Dz,
along the fibre axis and thecoupled rotation angles mA 5 mB are
shown in Figure6. Figure 6a is a representation of interchain
energy inrelation to coupled variations of mA and mB, with
theperpendicular offset Dx. Figure 6b shows the inter-chain
potential energy map at the optimum perpen-dicular offset Dx, as a
function of the translation Dz,along the chain direction and
coupled rotations of mA
FIGURE 6 (a) Interchain potential energy surface as a function
of coupled variation of mA and mBwith the perpendicular offset Dx.
Contours are drawn at intervals of 5 kcal/mol/cellobiose;
(b)Interchain potential energy map at the optimum perpendicular
offset Dx, as a function of thetranslation Dz along the chain
direction and coupled rotations of mA and mB. Contours are drawnat
5 kcal/mol/cellobiose intervals.
348 Vietor et al.
-
and mB. In this example, all primary hydroxyl groupsof the
cellulose chain are tg. This map exhibits anobvious symmetry; four
equivalent orientations (m,Dz), (m 1 1808, Dz), (m 1 1808, c 2 Dz)
and (m,c 2 Dz) are observed due to the helical nature of
thecellulose chain used as the model. Consequently, onlyone section
needs to be described. The most stablearea of the map, as
delineated by the first contourcovers a 20 angular range in m and a
0.15 nm rangein Dx. This indicates that there are multiple
interchainarrangements and that libration motion within thoselimits
could occur. Domains are found, which corre-spond to a somewhat
restricted range of mA. Verticaldisplacements are 0.00 or 0.279 nm
(slightly morethat h/ 2 5 2.59 nm); Dx appears to be
mainlydependent on mA, and much less on Dz. Within thesedomains, 17
energy minima were found within adifference energy window of 2
kcal/mol/cellobiose.Characteristics of the most energetically
stable ar-rangements are given in Table II. These arrangementshave
the chains stacked with ring surfaces touching.
Three-chain Arrangements. Three-chain arrange-ments were
calculated starting from each of the stabletwo-chain arrangements
calculated, by combining thechain duplex with a third identical
chain. Only stablearrangements with low energy were retained. In
mostcases, the global minimum for the three-chain
searches gave arrangements with the three chains inone plane.
Since such arrangements do not correspondto space-filling lattices,
they were excluded. As anexample, Figure 7 shows the energy and
lateral dis-placement as functions of m and Dz using a singlechain
combined with a chain duplex. The presence ofa pair of chains
instead of a single chain removes anumber of equivalent positions
that were visible inFigure 6. Here again, the most stable areas
extend inboth directions, suggesting that there are many
chainarrangements with comparable energy.
With the freely oriented chains, differences inchain orientation
(mA 2 mB) of more than ca. 10;caused the appearance of channels in
the corre-sponding lattice, so decreasing the chainchain
inter-action and the lattice density. Despite their impor-tance in
the description of the amorphous structures,such arrangements were
not suitable for generatingthree-dimensional crystalline
arrangements. Theywere therefore discarded.
Packing of TranslationalCellulose Chains
Two-chain Arrangements. Stable two-chain andthree-chain
arrangements were determined and se-lected as described for the
helical chains. Since thetranslational chains do not contain an
internal 21 axis;
Table II Optimum Values for the ChainChain Interactions for
Coupled Rotations of Helical Cellulose Chains
vmA()
Dz(nm)
Dx(nm)
E(vdW)(kcal/mol/dis)
E(HB) (kcal/mol/dis)
E(Tot) (kcal/mol/dis) Contacts
gg 277 0.00 0.481 27.24 0 27.24 154gg 277 0.520 0.488 26.42 0
26.42 145gg 285 0.226 0.501 25.50 0 25.50 132gg 286 0.219 0.503
25.47 0 25.47 131tg 277 0.00 0.482 27.03 0 27.03 159tg 260 0.274
0.468 26.17 0 26.17 150tg 296 0.324 0.544 26.89 0 25.89 135tg 258
0.000 0.477 25.60 0 25.60 120tg 288 0.270 0.522 25.18 0 25.18 135tg
274 0.520 0.494 25.07 0 25.07 127tg 250 0.179 0.487 24.97 0 24.97
130tg 2111 0.000 0.635 24.91 0 24.91 94tg 29 0.269 0.642 24.57 0
24.57 100gt 2126 0.279 0.669 25.61 21.95 27.56 101gt 2169 0.000
0.809 23.35 23.95 27.30 62gt 277 0.000 0.480 26.11 0 26.11 140gt
2101 0.219 0.550 25.68 0 25.68 139gt 261 0.279 0.468 25.65 0 25.65
138gt 277 0.520 0.488 25.39 0 25.39 138gt 258 0.00 0.477 25.13 0
25.13 114gt 288 0.209 0.508 25.08 0 25.08 119gt 29 0.264 0.646
24.76 0 24.76 100
Crystal Structure Prediction 349
-
only coupled rotations needed to be considered. Hy-drogen
bonding was not taken into account at thisstage. For the stable
configurations the energies foundwere higher than for the helical
chains. Also, thedistance between the chains was larger, and the
num-ber of chainchain contacts lower.
Three-chain Arrangements. No stable 3-chain ar-rangements that
resulted in a viable spacefilling lattice
compatible with a monoclinic lattice could be deter-mined.
Though several arrangements compatible witha triclinic lattice were
found, though these resulted inrather large unit-cell volumes
compared to those ob-tained for the helical chains.
Final SelectionUnit-cell parameters were calculated for the
selectedthree-chain arrangement obtained, allowing a space-
FIGURE 7 (a) Chain : chain-pair potential energy surface as a
function of chain rotation m andthe perpendicular offset Dx.
Contours are drawn at 5 kcal/mol/cellobiose intervals above the
globalminimum; (b) Chain : chain-pair potential energy map at the
optimum perpendicular offset Dx, asa function of the translation Dz
along the chain direction, and coupled chain rotations m.
Contoursare drawn at 5 kcal/mol/cellobiose intervals above the
global minimum.
350 Vietor et al.
-
filling lattice to be generated. The structures resultingin the
lowest cell volumes were retained. This gavetwo models for the
triclinic phase: one with helicalchains and one with translational
chains. The triclinicmodel with helical chains was considered to be
thepreferred candidate in view of its smaller unit cellvolume. Only
a model with helical chains could beretained for the monoclinic
phase; the translationalchains did not give acceptable three-chain
configura-tions.
The cellulose chains of both models are arrangedin sheets that
can be stabilized by hydrogen bonding.For the triclinic model,
hydrogen bonding betweensheets is not possible, whereas in the
monoclinicmodel one hydrogen bond per cellobiose unit
canparticipate in an intersheet link.
Description of the Selected Models
Two models were selected as representative of thecrystalline
structure of native cellulose. For both mod-els, the cellulose
chains were taken as parallel to thec axis of the unit cell.
Triclinic Model. The triclinic unit cell, space groupP1,
contains one cellobiose unit, and is shown inFigure 8. Unit-cell
parameters are a 5 0.63, b5 0.69, c 5 1.036 nm, a 5 113.0, b 5
121.1, g5 76.0. As the chains are related by translationalsymmetry
only, all have the same orientation aboutthe chain axis. The chains
of this structure are organ-ised into sheets oriented along the (1,
21, 0) plane.Within the sheets, the relative positions of the
chainsare constant, with a distance between helix axes of0.82 nm
and a displacement of 0.055 nm along thechain axis. These sheets
are stabilized by an inter-chain hydrogen bond between the OH6 of
one chainand the OH3 of the closest neighbor within the sheet(dOOO
5 0.30 nm). The distance between the sheetsis 0.433 nm, with a
displacement of 0.279 nm alongthe chain axis. No hydrogen bonding
was evidentbetween chains located in different sheets. An
intra-chain hydrogen bond can be deduced between the tgO6 and O2 in
the next ring along the chain (dOOO5 0.27).
Monoclinic Model. The predicted monoclinic ar-rangement
corresponds to space group P21 and con-tains 2 cellobiose units, as
shown in Figure 9. The unitcell parameters are a 5 0.87, b 5 0.75,
c 5 1.036nm, g 5 94.1. The chain axes coincide with symme-try axes
of the unit cell with one chain placed at thecorner of the unit
cell and the other at the center. Asin the triclinic lattice, the
chains are organized intosheets. In the present model, these sheets
are arranged
parallel to the bc (1, 0, 0) plane with a distance of0.435 nm
between the sheets. Two types of sheets canbe distinguished based
on a 5 difference in the ori-entation of the constituent cellulose
chains. Interchainhydrogen bonds are possible between O6 of a
selectedchain and the O2 of the closest neighboring chain(dOOO 5
0.250.28 nm, depending on the sheet) andnot O3 as in the triclinic
case. A further hydrogenbond may occur between the O6 of a chain in
onesheet and the O4 of the closest chain in a neighboringsheet
(dOOO 5 0.250.32 nm). O6 of each unitexhibits a gt orientation and
lies at 0.31 nm from O3of the next pyranose ring along the chain.
The differ-ence between our model and some suggested one forthe
monoclinic phase lies in the orientation of the O6primary hydroxyl
group. According to nmr measure-ment of the C6 chemical shift a tg
conformation hasbeen proposed as derived from the use of an
empirical
FIGURE 8 The triclinic model of cellulose I, viewed (a)along the
chain axis and (b) perpendicular to the chain axis.
Crystal Structure Prediction 351
-
correlation between chemical shifts and conformationof the
primary hydroxyl groups. As a consequence,intersheets hydrogen
bonds cannot be formed. Ofcourse, our calculations do reveal some
possible mon-oclinic arrangements for cellulose chains having
theirprimary hydroxyl groups in the tg orientation. How-ever, the
corresponding cell dimensions differ sub-stantially than those
experimentally derived. Beside,the energy of such arrangements is
slightly higherthan the one corresponding to our selected
model.
A monoclinic model has been recently proposed.34It is based on
the refinement of two independent setsof x-ray fiber diffraction
data. The comparison be-tween their proposed models and ours shows
a verysatisfactory agreement. The unit-cell parameters andthe space
group symmetry, along with the chain con-formation, are the same
except for the conformationof the hydroxymethyl group. A small
difference lies
in the relative orientation of the two chains. Thevariation of
the cylindrical polar angle between thetwo chains has been reported
to be either null orslightly negative (23.5) whereas the magnitude
ofthis angle is 5 in our predicted three-dimensionalstructure.
However, it was shown that the variationsof this angle in the range
210 to 10 have only aminor influence on the calculated agreement
factor.
Not surprisingly, the calculated cell dimensionsexhibit some
discrepancies with respect to those thathave been determined
experimentally. In the case ofthe monoclinic allomorph, the largest
deviationsamount to 0.03 nm and to 2.2. When compared to theaverage
experimental data, the maximum deviationsin the unit-cell
dimensions amount to 6% in lengthsand to 7% in the angles for the
triclinic model, and 8%in the lengths and only 3% in the angles for
themonoclinic model. Those variations reflect the occur-rence of
some flexibility in the interchain arrange-ments as revealed in the
present study. They alsoindicate that while our modeling protocol
providessatisfactory models, there is still room for improve-ment
in the methodology. Also, taking into accountthe possible
conformational changes that the hydroxy-methyl pendant group may
undergo, along with theoccurrence of some possible variations in
the pucker-ing parameters of the pyranose rings, might improvethe
end results in terms of the cell dimensions. Itshould nevertheless
be pointed out that in our simu-lation protocol yield prediction of
both the spacegroup symmetry and unit-cell dimensions, withoutany
other constraint than the polysaccharide fiberrepeat. It is
therefore remarkable to reach such anagreement for the dimensions
of the unit cells.
Lattice Energy CalculationsAs a final refinement, seven-chain
minicrystals weregenerated based on the unit cells obtained that
had thelowest cell volume. The conformational energy wasdetermined
after minimisation with MM3. Lattice en-ergy was determined as the
sum of all intermolecularenergy contributions involving the central
cellobioseunit of the minicrystal.
The results of these calculations are shown in Ta-ble III. The
structures based on helical chains showedonly minimal deformation
after minicrystal optimiza-tion. The structure based on a
translational chain, onthe other hand, was severely deformed toward
a sym-metrical helical chain. As a result of this deformation,the
lattice energy could not be usefully estimated.
Allomorphic TransitionsComparison of the triclinic and
monoclinic modelsindicates that the interchain distances are quite
simi-
FIGURE 9 The monoclinic model of cellulose I, viewed(a) along
the chain axis and (b) perpendicular to the chainaxis.
352 Vietor et al.
-
lar. Superposition of the two structures suggests that
atransition between them would not give rise to grosscrystal
deformations.
A transition from the triclinic to the monoclinicform would
require the following longitudinal shiftsand small rotations of
some cellulosic chains:
1. Reorient the conformation of OH-6 from tg togt.
2. Rotate the chains in every second sheet by 5about c.
3. Vertically shift every fourth sheet by 0.518 nm,i.e., c/ 2,
or a 180 rotation of the chains in thissheet (these changes are
equivalent due to thetwofold helical axis).
4. As a consequence of these reorientation, a ver-tical shift of
the third layer will result.
Given the absence of hydrogen bonding betweenthe sheets in the
triclinic model, the vertical shiftsalong the c axis should be
relatively accessible atelevated temperature. The formation of
intersheet hy-drogen bonds would then stabilize the structure
byfixing the chains in place. No lateral shifts would berequired to
facilitate the transition so leaving thelayers intact.
CONCLUSION
The present work has established a computationalprocedure to
predict the different ways that a poly-saccharide chain of known
conformation is able tointeract with other chain-like molecules.
The proce-dure has been applied to cellulose, for which
stableparallel chain pairings have been generated.
Few of these arrangements are capable of generat-ing an
efficiently packed three-dimensional array, but
may be pertinent to situations such as the amorphousstate or at
the surface of cellulose crystalline domains.
Structures with parallel chains were used for com-parison with
experimentally derived data. Theyshould provide sound starting
models for refinementagainst observed structure factors derived
from x-rayor electron diffraction, but should not be consideredas
complete descriptions of the two allomorphs ofnative cellulose.
Agreement between the predictedunit cell dimensions and the
published dimensions hasprovided some degree of validation of the
methodol-ogy.
The two most favorable predicted crystalline ar-rangements
correspond to a triclinic lattice, spacegroup P1, and to a
monoclinic form, space group P21.These structures correspond
closely to those whichhave been reported for cellulose Ia and Ib,
respec-tively. The cellulose chains in the selected modelsform
layers, stabilized by interchain hydrogen bonds.Stacking of the
layers to gives rise to the completecrystal lattice. Layer stacking
in the triclinic model isstabilized only by van der Waals
interactions. For themonoclinic model, the layers are linked
through twointerplane hydrogen bonds per cellobiose unit, one
toeach neighbouring layer.
The present algorithm is limited by its two-stagedetermination
of stable three-chain configurations,and acceptance of minimal
interchain distances whereall three chains are not in contact. A
new algorithmthat directly determines all stable three-chain
interac-tions is being developped in our laboratory.
The authors gratefully acknowledge financial support fromINRA to
RJV. The work was also conducted within theframework of CARENET-2 a
European funded networkwithin the Training and Mobility of
Researchers 19941998(MTL). The provision for financial support by
INRA andCNRS is acknowledged.
Table III Chain Conformation Parameters after MM3 Seven-Chain
Minicrystal Minimization (see Table I forOriginal Values)
Lattice t ()F() C ()
v()
v9()
c
(nm)
LatticeEnergy(kJ/mol)
Changea(nm)
TriclinicHelix 116 292 2149 170 169 1.036 220.0
0.0114Translational 115.3 285 2136 1.036 NDb 0.0562
MonoclinicHelix chain A 116.1 291 2157 63 61 1.036 215.7
0.0136Helix chain B 115.2 292 2149 66 66 1.036 218.0 0.0095
a Root mean square of displacement of the non-hydrogen atoms of
the central cellobiose unit (except OO6 and OO69) after fitting to
theoriginal conformation.
b Lattice energy could not be determined due to the large
lattice deformation.
Crystal Structure Prediction 353
-
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