1
An intermembrane ion trap
D.A. Morton-Blake*
School of Chemistry, Trinity College, Dublin 2, Ireland
Abstract
Molecular ion channels consisting of covalently stacked crown ether (CE) rings are
mounted in a membrane bilayer between two ion-containing aqueous layers. Molecular
dynamics shows that due to a large negative electrostatic potential on the axis of the
channel, the cations (but not anions) in the electrolyte layers enter the channel.
Calculations suggest that the cations encounter an energy barrier at the mouth of the
channel and occupy axial sites between which they may undergo a dynamic
redistribution. The application of electric fields sweep the ions to one end of the
channel but fields of electrolytic strengths effect neither the exit of the cations nor the
admission of anions. The retention of the ions M+ show that the CE channel acts as a
trap for cations from the electrolyte layers.
Key words: Passive transport, Ion migration, Ion channel, Molecular dynamics,
Electrostatic potentials, Solvation.
1. Introduction
The passage of ions across membranes is a central feature in the functioning of living
cells. Exciting developments in the manufacture and studies of molecular components
of the synthetic nanoworld [1] demand an understanding of how ions could also be
conveyed through artificial channels [2]. In biosystems the processes usually involve
voltage-gated channels and ionophores comprising proteins embedded in the
membrane [3]. The process involves conformational changes in the protein chains,
resulting in an active transport event. Although the transport mechanisms are known
and have been atomistically simulated for many of these processes [4] they are at
present too complex to use for the purpose of designing synthetic systems.
In the
work described here we atomistically simulate the passive transmembrane transport of
ions through a proposed synthetic channel.
Consider a two-dimensional lattice of molecular channels, each
molecule of which comprises a set of fourteen stacked 15-crown-ether-5 (CE) rings
which are mutually bonded by -NH- units [5] (Fig. 1).
In a later section of this work
Fig. 1
the channels will be embedded in a bilayer membrane of two C12H23COOH alkanoic
acid monolayers with each end in an aqueous electrolyte layer so as to form a
*
email: tblake@tcd.ie
telephone: 353 18961943
2
sandwich of two solutions separated by the membrane. The 40Å-long channels would
thus provide possible migration paths for the cation and anion species between the
aqueous layers.
2. Atomistic parameters
Bond lengths and bond angles in the carboxylic acid, the amine groups linking
adjacent CE rings and in the alkanoic acid chains, were the same as those used in an
earlier work [5] and were taken from literature values of diffraction and computed
quantities on model compounds [6].
In view of the importance of coulombic effects, which will be shown in this
work, a particular concern was the set of partial atomic charges, particularly those on
the CE ring atoms. These were obtained from Hartree-Fock calculations using the
DL_POLY package [7] either with or without Møller-Plesset corrections for electron
correlation (depending on the sizes of the molecules used for modelling).
The latter
compounds consisted of ethers, amines and alkyl-chain fatty acids. The values of the
oxygen charges q(O) in crown ether rings and model ether compounds were found to
vary significantly with the method selected to compute them.
Increasing the basis
set usually leads to higher negative charges: a Mullikan calculation of q(O) at MP2
level using a 6-311G basis set extended to d and f functions gave charges q(O) 
0.72e, but those calculated by other definitions of partial atom charges on a similar
level  Merz-Kollman-Singh scheme [8], Natural Population Analysis [9], and the
Breneman scheme [10]  were significantly less negative (0.3 to 0.4e). We
assigned a value of 0.60e to the CE oxygen atoms. The charge neutrality of the
HCOCH segment was conserved by placing compensating positive charges of
q(C) = +½q(O) to the adjacent C atoms and zero charges to the H atoms of the
segment.
The five electronegative oxygen atoms in each crown ether (CE) ring will
associate with an axially migrating cation M+. While this may be less likely to occur
for an anion migrant A a study of the electrostatics of the system in the next section
will note the effects of the partial positive charges on the ten carbon atoms which
adjoin the five oxygen atoms of the CE ring.
The molecular dynamics (MD) calculations were performed using DL_POLY
Smith and Forester [11] together with custom-written programs for data processing.
3
The simulation cell contains the molecular units defined in the previous section,
consisting of twenty-five 645-atom channels embedded in a bilayer composed of 1750
C12H23COOH molecules whose carboxylic ends were in contact with layers of
aqueous M+A solutions containing 7000 to 8000 water molecules and a varying
number of ions.
The atomistic potentials used for the non-bonded atom pair interactions
were DREIDING (6, 12) potentials [12] except those for the ions, for which those of
Åquist [13] were used for Li+, Na+ and K+ and those of Lee and Rasaiah [14] were
employed for F. Water molecules were simulated as Berendsen’s SPC model [15].
Potential functions for heteroatomic pairs were calculated by the geometric mean rule.
DREIDING
values were used for all A-B-C atom triads to define the energies
associated with three-atom ‘bond bending’.
No four-atom torsional potentials, such
as those describing bond torsions, were deployed.
3. Idealized ion channels
The molecular dynamics calculations, which will be described in subsequent
sections of the report, will be for the CE-channel structure defined above.
But if
synthetic ion channels ion channel systems are to be designed it would be useful first
to find optimisable features in a simple model that exercise a fine control over the
passage of the ions. As a prelude to the work, therefore, we examine the extent to
which the electrostatic potential along the axis of the channel governs the channel
migration of the ions.
This will be done by considering (1) a tube of n thin
concentric charged shells or hollow tubes each of different diameters but equal
lengths to represent the atom charges and (2) the electrostatics of a rigid single
channel of linked CE rings.
(1) Charged shells
Fig. 2
Fig. 2 depicts the concentric tubes (‘shells’) in the model used.
The ith shell
has radius ri and charge Qi and the channel has zero overall charge (  Qi  0 ). The
i
electrostatic potential dV at a point P distance p from the centre of a thin ring of
hickness dl with radius r bearing a charge dQ (Fig. 2a) is [16]
4
dV ( x) 
1
40
1
p2  r 2
where  is the surface charge density.
.dQ 
1
40
1
p2  r 2
. 2rdl
If the ring is at a distance l from one end of a
charged cylindrical shell then the potential at a point P at a distance p from the end
(Fig. 2b) is
V 
r
2 0

dl
(l  p) 2  r 2
Integrated over the cylinder’s surface from l = 0 to l = L, we have
r 1  Yl
V ( p) 
ln
2 0 1  Yl
l L
(1)
l 0
pl 
1
Yl  t a n t a n1

r 
2
where
(2)
Parameters
In the present molecular channel the innermost shell models the five oxygen
atoms in each crown ether ring at a radius r1 = 2.0 Å. The next shell, with r2 = 3.0 Å,
simulates the ten carbons C1 in Fig. 1 which flank the oxygen atoms of the first shell.
The 3rd shell contains the ten carbon atoms C2 at r3 = 3.8 Å that are bonded in pairs to
the five nitrogen atoms N in the 4th shell, for which r4 = 4.6 Å. The outermost shell,
with radius r5 =5.6 Å, represents the five amine hydrogen atoms H1.
Surface charge density values {i} were based on a comparison of partial
atomic charges calculated by the quantum chemical methods on model molecules as
described in the previous section.
Selected charges were varied in a manner that
preserved the ratios of the calculated charges of the atoms in the different shells.
Consequently the negatively charged five oxygen atoms per ring in the 1st shell were
taken to be compensated by the ten equivalent positively charged C1 atoms in the 2nd
shell according to q(O):q(C1) = 2 : 1 and the charges on the five amine units
C2NH1C2 per ring in shells 3 to 5 were distributed as q(N):q(C2):q(H1) = 5 : 2 :
1. The remaining atoms (the hydrogens bonded to the amine C2 atoms) were deemed
to be uncharged.
(2) Atomistic charges
5
We now represent the atoms by their electrical charges whose distributions are
governed by the same procedure as in the charged shell model. Each CE ring
therefore consists of five charges q(O) on the oxygen atoms, ten q(C1) on the C1 lying
on either side of these, five q(N), ten q(C2) and five q(H) on the on the amino groups
that interlink the CE rings.
This defines the axial electrostatic potential at a distance
p from the end ring as
14
5
V ( p)  
i 1 j 1
qj
40 r j2  ( z i  p) 2
(3)
where rj is the distance of charge qj from the axis in each CE ring and zi is the axial
position of the ith ring.
Results
Eqs. (1) and (2) were used to calculate the electrostatic potentials V(p) along
the axis of the channel fixing q(O) = 0.60e on shell 2 and q(C) = +0.30e on shell 1
for various values of q(N) on shell 4 which in turn determined the charges on shells 3
Fig. 3
and 5. Parts (a) and (b) of Fig. 3 compare the values obtained for the potential along
the channel axis obtained respectively by the shell and by the atomistic model for q(O)
= 0.60e and with different charges on the outer rings that relate to the amino
CNHC unit. The vertical lines on the horizontal axis at 0 and 40 indicate the
positions of the ends of the channel that is being simulated.
The variations of the
electrostatic potential are similar for the two models, particularly for low qN.
The role of the charges on the amino group that interlink the CE rings is clear
from the Figure: the more negative the charge on the N atom, the less negative is the
axial potential.
This is because a negatively charged N induces positive charges on
its adjacent carbon neighbours C2 according to the q(C) =  52 q(N) rule assumed in
the preceding section and the C2 atoms, with their smaller shell radius, are closer to
the axis (Fig. 1a). We shall later refer to the finding that for the more negative charges
on the amino unit’s N atom the potential is lower at the ends of the channel that at the
centre.
Consider a 1M aqueous solution of a 1:1 binary electrolyte M+A. Taking into
account the hydration of M+ in aqueous solution, if the cation’s closest approach to an
ion A is about 2.0Å, it would experience an electrostatic potential q/40r of about
6
7v at this point. The dielectric water molecules cause this negative potential to
increase to zero within a Debye length D (in Å) of about 3.04/cM for an aqueous
electrolyte solution of molarity cM [17], or over a distance of a few Å. The rise in
potential over this distance with separation from A, followed by its subsequent drop
to the ca. 10v values on entry to the channel (shown by the electrostatics leading to
Fig. 3) indicates that an energy barrier must be overcome by M+ before it can be
transported through the low-potential environment of the molecular channel.
4. Molecular dynamics investigations
4.1 Simple lattice of molecular channels (no membrane or water)
This sub-section examines the channel motion of the ions in the absence of the
bilayer membrane and the aqueous layers.
With H atoms terminating the end CE
rings (atoms rather than the COOH groups used in the membrane calculations), the
length of this channel is 29.5Å.
The migrating ions are far less numerous than the remainder of the system to
be simulated; the statistics of their dynamics in the molecular channel are therefore
subject to uncertainties arising from the paucity of data. We counter this by defining
the MD cell to contain multiple copies of the channel and migrants. A square 5  5
two-dimensional lattice is assembled from twenty-five 14-ring channels of the
Fig. 4
kind illustrated by Fig. 4.
A partial charge of 0.65e was accorded to the O atoms in
the CE rings and 0.60e to the N atoms of the inter-ring amino groups. In the absence
of a bilayer membrane in this part of the work we prohibited the motion of the
channels by fixing the coordinates of one atom at each end-ring. Twenty-five
monopositive and an equal number of mononegative ions M+ and A were then
assembled near the ends of each channel, first removing the symmetry equivalence of
the migration systems by imposing a slight degree of disorder on the initial positions
of the ions.
But by simply placing all the M+ ions at one end of the channels and all
the A at the other ends a ‘ferroelectric’ imbalance would be created. This was
avoided by making an alternating ‘head-to-foot’ arrangement of the M+ and A ions
so that the electric dipoles change their directions over successive channels, resulting
in zero macroscopic dipole moment for the overall simulation cell.
7
Results
With the twenty-five Li+ ions initially fixed at their sites near the channel ends,
the rest of the lattice was allowed to relax fully.
when they are subsequently released.
Fig. 5 shows the ions’ trajectories
In a brief time interval that occurs before the
new lattice system has thermally equilibrated, the ions enter the channel and the
regular intervals between their axial positions indicate that they take up positions that
correspond to the periodicity of the molecular channel. Fig. 5a records the amplitudes
and regularity of their fluctuation at these positions over the first 2000 time steps (2 ps)
after thermal equilibrium is deemed to have been established. The Figure shows that
the ions undergo periodic dynamical motions between sites with atom ‘snapshot’ plots
indicating that these positions are within the channel cavities between the planes of
the CE rings. In the course of a longer time interval of
Fig. 5
600,000 time steps (0.6 nanoseconds) in Fig. 5b, several transitions between the
channel sites are observable. Another feature that is illustrated in the Figure is the
greater occupation of channel sites near the ends of the channels (a larger number of
traces occur at the top and at the bottom of part (b) than in other parts).
We
tentatively associate this result with the feature illustrated by those curves in Fig. 3b
that were obtained when more negative charges were placed on the N atoms.
It was
commented at the end of section 3 that these traces indicated that the electrostatic
potential was more negative near the ends of the channel than at its centre.
4.2 Other ions
Fig. 6
Fig. 6 shows a comparison of the trajectories obtained for Li+, Na+ and K+
from the moment that the cations were released, without permitting them time to
thermally equilibrate. They illustrate how the ions (particularly the heavier two),
confined by the energy barriers just outside the channel (Section 3), initially undergo
‘explore’ the resulting cavity before taking up their final sites. The oscillations
support the importance of the coulombic forces describing the behaviour of the ions in
the channel as they are not immediately offset by the steric inhibitions of the ions
passing through the CE rings.
8
Similar procedures to investigate the negatively charged ions (F, Cl, )
found that the anions did not enter the channel. As the ionic radius of the smallest of
these – the fluoride ion – is only 1.33 Å while that of the K+ ion (which does migrate
in the channel) is slightly larger at 1.38 Å [18] we may again conclude that the
passage of the anions is determined principally by electrostatic, rather than steric,
considerations.
4.3 Membrane-embedded CE channels
Fig. 7
In order to study the transport of ions between electrolyte solutions across a
membrane, the CE molecular channels are embedded in an amphiphilic bilayer
membrane separating the two solutions (Fig. 7). The bilayer membrane is
constructed as two sets of aliphatic-tailed carboxylic acid molecules C12H23COOH.
In each set the carboxylic COOH units are in the ab plane of the orthogonal MD cell;
the molecules are placed tail-to-tail (–CH3H3C–) along the cell’s c axis. The
COOH ‘heads’ in each layer are in hydrophilic contact with the external aqueous
layers.
In order to confer the necessary hydrophilicity on the CE channel’s two end
rings which are in contact with the polar electrolyte solutions, five of the ten carbon
atoms in each of these rings were provided with electrophilic COOH substituents.
After ensuring that there were ions in the electrolyte solutions near the ‘lower’
25 ends of the channels, the positions of these ions were fixed while the remainder of
the atomistic system was thermally relaxed. The MD was then run with the ions
released for the same time interval as those in the simple channel lattice in Section 3.1,
where the Li+ ions had migrated through a ‘bare’ lattice of 5  5 CE channels that
were not in contact with aqueous solutions. When the trajectories for the ions in the
Fig. 8
5  5 lattice of [channels + bilayer membrane + aqueous layers] in Fig. 8 are
compared with those in Fig. 6a for the bare lattice an important feature is apparent.
In the absence of the aqueous layers we saw in Fig. 6a that the Li+ ions, effectively a
gaseous phase system, reached their favoured channel lattice sites in a very short time
interval of 1000 time steps (1 ps); by the end of the 10000 ts run, all 25 ions had
travelled into the channels and many of the sites had been occupied. In contrast, Fig.
9
8 shows that for the membrane-aqueous lattice in the same interval only 9 ions have
now entered the channel. We may interpret this by saying that when the Li+ ions are
initially in contact with the molecules of the aqueous layers they are solvated so that
their migration into the channel is inhibited. In order to be released from their
solvated condition the ions must overcome the energy barrier, which was described at
the end of Section 3 to be due to a region of high electrostatic potential near the
mouth of the channel. The cation’s stability, restored by the marked potential drop
near the end CE ring, ensures that once they have entered, the Li+ ions are retained
and do not drift back out of the channel.
Other cations (Na+ and K+) behaved similarly to Li+, but as expected from
remarks made in the previous sub-section we again found that due to the numerically
large negative axial potential, F anions failed to enter the channel.
The positions shown to be accessed by the Li+ ions in Fig. 8 are always within
the channel; MD runs conducted for up to 106 time steps showed no ions exiting. The
fact that the Li+ ions are admitted to the channel by a diffusive (passive) migration but
precluded from subsequently leaving it implies that the channel is acting as a cation
trap. No water molecules were found to have been electrophoretically ‘dragged’ into
the channel by the Li+ ions. In the region near the channel mouth an energy barrier is
created as the cations are stripped of their hydration.
Fig. 9
In efforts to allow the ions to penetrate the channel and subsequently to exit
into the second aqueous layer, Fig. 9 displays the results of applying electric fields of
intensities 0.1, 0.2 and 0.4 v Å1 along the channel axes. The Figure shows that the
action of the field is to sweep the ions into the channel, but not out of it. The
intensities of the electric fields applied are of the orders of those found on molecular
scales but would result in dielectric breakdown if applied in a laboratory environment.
We therefore conclude that inducing the migration of ions through the channel by
applying external fields of ‘electrolytic’ strengths is not possible.
It might be thought that electric fields that drive the Li+ ions into the channel
might do the same for the F ions which were at the opposite ends of the channel from
those of the Li+ ions. However molecular snapshots showed that electric fields of up
to 1 v Å1 resulted in the admission of F anions to a distance of about 1.0 Å into two
of the 25 channels in the system, but even after 106 time steps they failed to proceed
10
any further. In other words, the presence of the positive electrostatic potential on the
channel axis, even for the strongest electric fields, did not permit F ions to proceed
beyond the COOH atoms on the channel’s first CE ring. In this region they formed an
anion layer that offset the overall positive charge that now characterised the channel
and its contents.
That the bare Li+ ions had indeed entered the molecular channels was
ascertained from atomistic ‘snapshots’ of the structure at various time-steps. These
pictures showed the ions to be clearly at various (mainly off-axial) positions in the
Fig. 10
channels, but no water molecules had entered. Fig. 10 contains a more concise
demonstration of the entry of the Li+ ions into the channel in the form of radial pair
distribution functions gLi-A (r) for atom pairs Li-A. The greater the amplitude of the
peak in the gAB (r) trace when A and B are separated by distance r, the higher the
probability of finding the atoms at that separation. The three parts of the Figure
show the traces of the radial distribution functions (rdf) for (a) the (Li+water), (b) the
(Li+O) and (c) the (Li+N) pairs. Each of (a), (b) and (c) consists of three traces,
referring to the Li+ ions in (1) the bulk aqueous solution, (2) when the ions are close
to the mouth of the channels and (3) after they had entered the channel. (This
investigation was enabled by declaring two Li+ species – one for the ions in the
aqueous layer, and another for the Li+ ions that were monitored as they entered the
channel. (Similarly, the atom at the ‘water’ end of the rdf pair in (a) was a separate
species from the oxygen of the channel rings in the (b) trace.) During the three-stage
entry of the Li+ ions into the channel, the sharp depletion and elimination of the (Li+ 
water) peak in part (a) of the Figure, and the enhanced peaks for (Li+  O) and (Li+ 
N) in (b) and (c) demonstrate the departure of the ions from the aqueous layers, the
shedding of their water atmosphere and their admission as bare cations into the CE
rings of the channel.
5. Conclusions and discussion
The electrostatic potential calculated by the procedures (1) and (2) in Section 3
led to consistent magnitudes of ca. 10v. As this quantity is for points in the
channel relative to the vacuum it does not directly relate to measured transmembrane
11
potential data. The latter have values (usually less than 1v) which are between points
in the aqueous media that are inside and outside the cell.
Unlike events in biosystems, the rigidity of our crown-ether channel prohibits
modifications in its secondary structure to respond to the presence of a migrant. Its
transporting character is consequently passive.
The five oxygen atoms, and to a
lesser extent the five nitrogens, in each CE unit produce the strongly negative
electrostatic potential along the axis channel. As a result the channel spontaneously
admits cations where, after oscillatory ‘explorations’ they occupy sites in the
molecular cavities between the CE rings, between which redistributions (‘transitions’)
can occur. The fact that similar behaviour is shown by Li+, Na+ and K+ ions, but not
by anions (F) is interpreted as the dominating role played by the electrostatic over
steric effects in the molecular system. This is because, despite the similar sizes of the
K+ and F ions, the latter species does not enter the channel, even with the inducement
of strong axial electric fields.
With no electric field applied to the aqueous/membrane/channel system the
entry of cations from the electrolyte layer to the channel is significantly slower than
from a ‘gas phase’ environment of ions.
This observation is consistent with the
proposed energy barrier that the M+ ions must overcome when they migrate from their
low-potential environment in which they are electrostatically bound to surrounding
water molecules (and to A anions) to another low-potential environment that is found
on the axis of the molecular channel.
In the aqueous layers there is a narrow region
between the bulk solution and the channel ends where the M+ ions are hydrated to a
greater extent on the aqueous side than on the channel side. (This space is
reminiscent of the well-known one containing solvent molecules near a phase
interface where their stabilizing interactions are less abundant near the interface than
on the opposite side, leaving the molecules in a condition of higher potential energy.)
The region marks that of a barrier in the energy profile describing the passage of the
ions from the bulk liquid layers into the channel. After crossing the barrier the
channel presents itself as a deep energy well from which strong electric fields cannot
expel the cations. As a result of the channel’s property to occlude these ions it should
be characterised as a cation trap rather than as a cation channel.
The system investigated in this work was one in which the CE channels are in
contact with two different environments; they are therefore amphiphiles, their exterior
12
atoms being selected to match the polarities of their surroundings. The end rings,
which were in aqueous layers, were made compatible with the water by giving them
hydrophilic carboxylic acid groups (COOH), while the H atoms on the CE rings in
the main body of the channel resulted in their hydrophobic consistency with the nonpolar bilayer membrane in which they were embedded.
In another structural
modification, containing no bilayer membrane, the channel could be made entirely
hydrophobic by H or alkyl-chain substituents.
In aqueous solutions of M+ A
electrolyte the insoluble channels would form a two-phase system. On mixing, the
cations would enter the channel, leaving the A outside but bound to the now
positively charged channel by electrostatic forces, thereby defining an agent that
desalinizes the electrolyte.
Work on this aspect of the channel system is in progress.
Acknowledgement
The computational facilities of An Institúd um Theicneolaíocht Eolais agus
Riomhfhorbairt na hÉireann (IITAC) were used to conduct this investigation.
13
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16
Figures
Fig. 1 (a) The basic crown ether unit and (b) the stacked components that comprise
the molecular ion-channel.
Fig. 2
The shell model of ion channel migration.
Fig. 3
A comparison of the axial electrostatic potential in the ion channel for the
two models considered for different charges corresponding to the N atom in ring 4.
In the shell model (a) the charge on the inner ring corresponds to a charge of 0.60e
on the CE oxygen atoms in the atomistic model (b).
Fig. 4
An ion channel of 14 CE rings linked by NH units.
Fig. 5
(a) Short- and (b) long- interval migration of Li+ ions through a lattice of CE
ion channels.
Fig. 6
The channel dynamics of (a) Li+, (b) Na+ and (c) K+. Plotting symbols
distinguish the 25 channels.
Fig. 7
ac-plane cut-away section through the lattice to reveal the 1st row of the 25
CE ion channels embedded in a C12H23COOH bilayer, separating two aqueous layers
containing a 1:1 electrolyte (top and bottom).
Fig. 8 The entry of Li+ ions into the CE channels when the channels are embedded
in a bilayer membrane between two aqueous layers. No external electric field is
applied. Compare the result with the spontaneous entry shown by Fig. 5(a).
Fig. 9 The migration of a Li+ ion through a bilayer lattice between aqueous
electrolyte layers in the presence of electric fields of different strengths E.
Fig. 10.
Pair radial distribution functions for Li+ with water, ring-O and ring-N
monitoring three stages of the ingress of the Li+ ions from the bulk aqueous solution
into the molecular channel. (  ) describes the Li+ ions in the bulk aqueous solution,
17
(----) is when the ions are close to the mouth of the channels and () is the rdf for
the Li+ ions after they have entered the channel and equilibrated.
18
(a)
(b)
Fig. 1
19
Fig. 2
20
(a)
(b)
Fig. 3
21
Fig. 4
22
ch 1
ch 2
ch 3
ch 4
ch 5
ch 6
ch 7
ch 8
ch 9
ch10
ch11
ch12
ch13
ch14
ch15
ch16
ch17
ch18
ch19
ch20
ch21
ch22
ch23
ch24
ch25
25
axial migration (Å)
20
15
10
5
0
0
500
1000
Timesteps (units fs)
(a)
(b)
Fig. 5
1500
2000
23
(a)
(b)
24
(c)
Fig. 6
25
Fig. 7
26
Fig. 8
27
Fig. 9
28
(a)
(b)
(c)
Fig. 10