In the Classroom
Nanotechnology Provides a New Perspective on Chemical Thermodynamics Richard G. Haverkamp Institute of Technology and Engineering, Massey University, Private Bag 11222, Palmerston North, New Zealand;
[email protected] By stretching a single polymer molecule it is possible to use a mechanical spring balance to measure thermodynamic properties of that molecule. Many of the thermodynamic properties of a system of molecules are apparent within a single polymer molecule. A single polymer molecule can act as its own ensemble with a distribution of energy between its parts and an equilibrium between different forms of these parts. A welldefined conformational change within certain types of polysaccharide molecules (rather than a collection of molecules) can be analogous to a chemical reaction with an equilibrium constant, Gibbs energy, enthalpy, and entropy applying to the individual molecule. These concepts can be used in undergraduate physical chemistry courses to fascinate students about new ways of understanding physical chemistry and of measuring thermodynamic properties that arise from the techniques of nanotechnology. Polysaccharides Polysaccharides are ubiquitous in living systems, for example, cellulose, pectin, and amylose in land plants; alginate and carrageenan in marine plants; chitin in crustacea, insects, and fungi; dextran from bacteria; and a variety of polysaccharides in animals including hyaluronan, dermochondan sulfate, and chondroitan sulfate. Polysaccharides (poly- from Greek polumeros, having many parts and ‑saccharide from Latin saccharum, sugar) may be composed of chains of six-membered sugar rings (5 carbon and 1 oxygen forming the rings, called pyranose
rings), each ring linked by an additional oxygen. These rings can be in the “chair” or “boat” form (Figure 1). The chair and boat forms can interconvert but have different energies and there is an energy barrier to conversion. Each carbon can form two bonds outside the ring. These bonds are defined as either equatorial, e, or axial, a (1). These sugar, or glycan, rings may have different substituents, for example hydroxyl, carboxyl, amine, and ether. These substituents influence the properties of the polysaccharides; however, the manner in which the glycan rings are linked, called glycosidic linkages, is also important. The linking oxygen may be attached to each of the glycans either axially or equatorially. For example some polysaccharides have two equatorial linkages (e–e) such as cellulose or chitosan (Figure 2A). Other polysaccharides may have all axial–equatorial (a–e) linkages such as dextran or amylose, while others have axial–axial (a–a) linkages such as pectin (Figure 2B). There are also many polysaccharides that have mixed linkages such as alginates (e–e, e–a, and a–a) or dermachondan sulfate (predominantly a–e but some e–e) (1). Pyranose-based polymers with a 1–4 glycosidic linkage with one axial and one equatorial bond, such as amylose (1a–4e) and dermochondan sulfate (1a–4e), can undergo one main conformational transformation. Those with a 1–4 linkage but with both bonds axial, such as pectin (1a–4a), undergo at least two types of conformational transformations. Conversely those with solely equatorial linkages, such as cellulose (1e–4e) do not have such possibilities. Single Molecule Stretching
O
O
Figure 1. Boat (left) and chair (right) forms of the pyranose ring.
A
COOH O
B
HO HO
OH
OH O
HO NH2
O HO
O NH2
O
O
COOH O
HO HO
O
Figure 2. Structure of two polysaccharides: (A) chitosan, 1e–4e and (B) pectin, 1a–4a.
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With the advent of the atomic force microscope (AFM) it has become possible to stretch individual molecules of polymers and obtain distance–force response curves (2–4). Many kinds of polymer molecules have been subjected to this investigation including DNA, proteins, synthetic polymers, and polysaccharides. The experiment is both simpler and more difficult than might be imagined: simpler in that it is necessary only to have a dilute solution of the polymer and more difficult in getting the correct conditions to grab a molecule with an AFM cantilever and stretch it sufficiently. These conditions can include getting the correct concentration, substrate, pH, salt concentration, and coating on the AFM tip. The cantilever must be carefully calibrated for its spring constant so that cantilever deflection can be interpreted as the force applied to the stretched molecule. A simple plot of the force with which a molecule resists the stretching by the cantilever is obtained from the experiment. This plot does not start at zero length since the likelihood of two ends of the molecule being beside each other (zero length) is low (5). For polysaccharides force curves can have different shapes. The shape of the curve depends on the glycosidic linkages. For e–e linked systems a simple curve (e.g., hyaluronan, Figure 3A) is observed. For e–a linked polysaccharides a curve
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In the Classroom
with one shoulder in it is observed (e.g., dextran, Figure 3B) and for a–a linked polysaccharides two shoulders may be observed (e.g., pectin, Figure 3C) (6). The inflection in the force–extension curve arises from an effect that is called a molecular lever (9). Each ring within the molecule can flip between the chair and boat forms. Pulling the
Force / nN
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Thermodynamics of Stretching Single Molecules
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Several aspects of chemical thermodynamics in stretching polysaccharide molecules of the clicking type can be illustrated.
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Maxwell–Boltzmann Distribution Just as there is a distribution of speeds of individual gas molecules in a jar of gas (which depends on temperature) so there is a distribution of energy between each of the glycan rings in a polysaccharide. This is a Maxwell–Boltzmann distribution, which can be expressed in the form: (1) Pi C e Ei / kB T where Pi is the probability of finding the gas molecule (or glycan ring) in the energy state Ei, kB is the Boltzmann constant, T is temperature, and C is a constant. The Boltzmann distribution of energy in states of a gas is well-known to students of thermodynamics and also applies to parts of a polymer molecule. In some respects, this is not a new concept as the Boltzmann distribution of energy between rotational, vibrational, and translational modes in a molecule forms the basis of statistical thermodynamics. However, the measurement of the distribution of energy between the glycan rings of the polysaccharide is intermediate between the considerations of individual energy modes for statistical thermodynamics and bulk systems for classical thermodynamics. The unit chosen for considering the energy distribution is the glycan ring, of which there may be 100–500 in a typical polysaccharide molecule. Equilibrium For a chemical reaction, such as A C, an equilibrium exists between the products and the reactants. The concentration of the reactants, A, and products, C, will approach the equilibrium concentrations (subject to kinetic restraints).
1.5
Force / nN
ends of the molecule will flip the glycan rings from the shorter form to the longer form. Each of these flips between chair and boat forms within the molecule is referred to as a click (Figure 4). The changes between forms under the influence of a force applied to the ends of the molecule form the basis for understanding and measuring the thermodynamics on the nanoscale discussed in this article.
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Extension / nm Figure 3. Force versus extension curves for (A) hyaluronan, 1e–4e, no clicks; (B) dextran, 1e–4a, one click at 33.1 kJ mol–1; and (C) pectin, 1a–4a, two clicks, 12 kJ mol–1 and 17 kJ mol–1 (6, 7).
1st transition 1
1
2nd transition
4
Figure 4. Chair-boat-chair transitions by the molecular lever (9).
< A>
For a single polysaccharide molecule, which may contain clicking rings, an equilibrium exists between clicked and unclicked states, unclicked clicked, and the equilibrium constant, K, determines the concentration of the clicked (longer) and unclicked (shorter) parts:
4
K
K
< unclicked >
(2)
This is a direct consequence of the Maxwell–Boltzmann distribution of energy throughout the polymer. Equilibrium is a dynamic state with exchange constantly occurring between the components on both sides of the equilibrium. Therefore, in a polysaccharide molecule containing
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In the Classroom
clickable rings there is a constant exchange taking place between clicked and unclicked parts of the molecule but with the ratio of the total number of each of these remaining approximately equal to the values prescribed by the equilibrium constant. Gibbs Energy The equilibrium constant, K, for a chemical reaction arises as a consequence of the Gibbs energy difference between the reactants and products, ΔG, with a Maxwell–Boltzmann distribution between the two states:
%G kB T ln K
(3)
Therefore a measurement of the Gibbs energy difference between the reactants and the products will specify the equilibrium concentrations of the reactants and products (or vice versa). For a polysaccharide molecule containing only one type of click the Gibbs energy difference between the molecule where no rings are clicked and where all rings are clicked determines the ratio of clicked, c, to unclicked, u, rings
% c u G kB T ln K
(4)
where Δ(c−u)G is the Gibbs energy difference between the completely clicked and the completely unclicked conformers (7). This is a Gibbs energy of a reaction within a polysaccharide molecule, rather than of a collection of molecules. Shifting the Equilibrium by Force Le Châtelier’s principle is a useful guide to equilibria: A perturbation of a system at equilibrium will cause the equilibrium position to change in such as way as to tend to remove the perturbation. For example, if a reaction proceeds with a positive volume change the application of pressure shifts the equilibrium in favor of the reactants. This can be expressed in thermodynamic terms
v%G vP
T
%V
(5)
where P is pressure, V is volume. In the case of a polysaccharide molecule containing only one type of click the application of a stretching force to the ends of the molecule will shift the equilibrium in favor of the clicked (longer conformer) states in an analogous way
v%G vF
T
% L
(6)
where F is the stretching force applied to the ends of the molecule and Δ L is the change in length of the molecule due to the combined effect of all the clicks (but excluding the components of elastic stretching and entropic straightening of the chain). Another way to view this is to consider the change in ΔG for the click under an applied force. This new Gibbs energy, ΔGF, will be a result of the modification of the Gibbs energy for the unstretched molecule, ΔG0, by the sum of the products of the force applied during each click, Fi , times the extra length contributed by each click, Δ li:
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%GF %G 0
¥ Fi % li
(7)
As more force is applied, this force changes the Gibbs energy of the conformational change, ΔGF, and therefore changes the equilibrium between the clicked and the unclicked conformers (7, 10). The concept that an internal molecular equilibrium can be altered by pulling on a molecule rather than changing some macroscopic property of the system is quite foreign to most chemistry students. Measurement of Gibbs Energy At constant temperature and pressure the Gibbs energy is defined by %G % H T %S (8) Traditionally the Gibbs energy for a chemical reaction is determined by measuring the enthalpy (ΔH) and the entropy (ΔS) (11). The enthalpy, or heat of reaction, can be determined in a calorimeter where the heat released (or absorbed) by combustion or chemical reaction is measured. The entropy can be measured by measuring the heat capacity (Cp) of the reactants and products for a bulk material according to T
S S0 0
Cp T
dT
(9)
Alternatively, for electrochemical reactions, ΔG can be obtained from the electrochemical potential, E,
%G n .E
(10)
where ℱ is Faraday constant and n is the number of electrons transferred in the reaction. In contrast, for the conformational change (sum of the clicks) in a single polysaccharide molecule it is not practical to measure the heat of reaction directly. However, by fitting a mathematical function to the force–extension curve, which is obtained mechanically, it is possible to extract this Gibbs energy directly (7). This is quite remarkable. A small mechanical device, measuring a force and the distance over which this force is applied, can be used on a single molecule to obtain a fundamental thermodynamic property, the Gibbs energy. Normally these thermodynamic properties for a substance would be obtained by measuring a heat of reaction and heat capacity and the measurements would be done on a large collection of molecules. It is now possible, with the new tools of nanotechnology, to do this measurement mechanically. Furthermore, normally the measurements would be done on a large collection of molecules whereas here the measurement is performed on just one molecule (although this consists of a reasonable sized collection of reaction units, the glycan rings). While statistical thermodynamics can link the vibrational and rotational modes within a molecule to the bulk thermodynamic properties of an assemblage of molecules, it is a predictive technique and not a measurement technique. However, single molecule stretching is a measurement technique and a fundamentally different way of measuring thermodynamic properties. It links, in a simple and remarkable way, the physics of springs and forces with chemical reactions and equilibria.
i
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F
kB T Lp
1 L F 1 ' 4 Lc
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L F 1 ' Lc 4
%G0 F %L Le Lp %L 1 exp kB T
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where Lp is the persistence length (a measure of the stiffness of the polymer), Φ is the modulus of elasticity (the “extensible” part of the model), Lc is the contour length (this is the length of the straightened but not stretched molecule), and the other terms are as defined previously. This can be used to fit a non-clicking polysaccharide, such as hyaluronan, and a routine can be written in a program such as Matlab to fit this equation to the data. For the more complex clicking polysaccharides this equation needs to be modified to accommodate the ring conformational transformation (7). This can be done by substituting Lc in eq 11 by a term referred to as Le , which incorporates the change in length due to the clicks. Starting with the length of the unclicked molecule, Lp, this value is increased by the extra length added owing to the rings that have clicked, ΔL:
A
Force / nN
Statistical Mechanics of Long Chain Molecules The behavior of long chain molecules, such as polysaccharides, proteins, and DNA, can be explained by statistical mechanical models. These models are described in detail in a number of textbooks (8, 12). The models of flexible polymer chains are based on the assumption of bending and coiling randomly and are referred to as random-coil models. There are a number of versions of these models that have constraints added to restrict the movement in the molecule. These constraints result largely from the rotational barriers of the skeletal bonds. As a simple example of the magnitude of this force, the energy difference to rotate around the central C−C bond in n-butane between trans and gauche conformations is about 2 kJ mol‒1 (8). At higher stretching forces in polymers the bond angles may change or the C−C skeletal bonds may begin to be slightly elongated. One of these models can be selected, and here the extensible worm-like chain (13) is used to fit data obtained from a stretching experiment. One formulation of the extensible worm-like chain is
1
(12)
This can then be used to fit stretching data for polysaccharides that click. Exercise Using the experimental data for the stretching of a dextran molecule (see the online material) the extraction of the Gibbs energy for the conformational transition in this molecule can be illustrated. The data are often recorded as force versus sample position or force versus cantilever-base position. The aim is to stretch the molecule at a constant rate, but this is not easy as a variable force on the cantilever causes the cantilever to bend by a variable amount. If the force experienced by the AFM tip
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Extension / nm Figure 5. Dextran single molecule stretch: (A) approximate midpoint of transition drawn by eye as a horizontal line at about 0.9 nN and (B) data with fitted curve.
is not constant, which it is not in these experiments, the tip of the cantilever does not move at a constant rate. The raw force–distance data (cantilever base movement not cantilever tip movement) therefore has to be adjusted for the deflection of the cantilever. The data presented in the online material has been adjusted for this. The data are plotted (Figure 3B) and the midpoint force for the transition can be estimated from the plot (Figure 5A). This appears to be ~900 pN. The data are then fitted to eq 11 using a fitting routine written in Matlab. It is helpful if the increase in length due to each click is known: for dextran Δ li = 0.065 nm (this can also be obtained by fitting). Because the number of rings in the molecule to be stretched is not yet known it is necessary to substitute all the L values (L, Le , Lp) by N multiplied by l (l, lp, le), where N is the number of rings in the molecule. N becomes a fitting parameter. The fit is illustrated in Figure 5B. From the fitting routine values for N, Lp, Φ, and, of most interest, ΔG0 for the conformational transformation are obtained. For the data provided for dextran, this fit gives ΔG0 = 33 kJ mol‒1. This value is now used to obtain the force at which half the rings will have clicked. This is found to be 843 pN, which is a little lower than the original eyeball estimate.
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Other Thermodynamic Properties from Stretching Single Molecules ∆H and ∆S for Single Polysaccharide Conformational Changes Traditionally to determine ΔG for a chemical reaction ΔH and ΔS are determined and then ΔG is obtained by the relationship in eq 8. To obtain ΔH, ΔS would normally require a measurement of the heat of reaction and the temperature dependence of the heat capacity. These are clearly not easy, or normally very meaningful, things to do on a single molecule. An alternative method of getting ΔH and ΔS for a reaction is to measure the temperature dependence of the equilibrium constant. The Gibbs–Helmholtz equation describes this relationship: %G v (13) %H T 2 vT T For a single polysaccharide molecule the Gibbs energy is measured directly by the force–extension curve. Therefore to obtain the enthalpy and entropy one more variation on the measurement is needed: a polysaccharide molecule is stretched at different temperatures. Then a plot of ΔG∙T versus T gives a slope of –ΔH∙T 2 to obtain the enthalpy. This is effectively a van’t Hoff plot (except that ΔG∙T is plotted rather than Rln Kp∙T). From the enthalpy and Gibbs energy the entropy is obtained (eq 8). The temperature dependence of the conformational transformation force is used to obtain the enthalpy and entropy of that structural change (14). Fluctuations from the Mean In these illustrations the similarities between thermodynamic aspects of a single molecule and a chemical reaction or gas-phase system have been demonstrated. However, in a typical polysaccharide there are only 100–500 entities compared with ~1023 for a chemical reaction. Therefore it might be expected occasionally to measure some deviations from the equilibrium state. Small variations in energy can be seen by the apparent “noise” in the force–extension curves. Where the slope of the curves is small, a small energy difference may produce a large displacement (in the cantilever and hence in the apparent force measurement); whereas where the slope is large, the same energy perturbation results in a smaller displacement (15, 16). Statistical fluctuations have also been found experimentally in single molecule stretching where for periods of up to one second the motions of parts of a molecule do not travel down the energy gradient, but may move against the thermodynamic gradient. Although at first glance this might appear to defy the first law of thermodynamics, of course it does not since thermodynamics is only perfectly applicable to infinitely large collections of objects. These are therefore only the predicted statistical fluctuations (17). Nevertheless, it is remarkable that these fluctuations can be observed. Conclusions Chemical thermodynamics, as taught in undergraduate courses, can also be viewed from the perspective of a single polysaccharide molecule that undergoes conformational changes. When the molecule is stretched mechanically, thermodynamic properties can be measured from the force–extension curve. These properties 54
include (i) energy distribution within the parts of the polymers; (ii) equilibrium within the molecule of chair and boat forms; (iii) Gibbs energy of conformational transitions (clicks) within the molecule; (iv) shifting the equilibrium by an applied force; (v) measurement of the Gibbs energy from a force–extension measurement; (vi) measurement of enthalpy and entropy by a force–extension measurement at different temperatures; and (vii) fluctuations against the thermodynamic gradient. This is a fundamentally different approach from measuring heat capacities as in classical thermodynamics or calculating from vibrational and rotational modes as in statistical thermodynamics. This can be used to illustrate how the new techniques of nanotechnology can provide an alternative route to well established thermodynamic principles. Acknowledgment The author would like to thank Aaron T. Marshall for assistance with the figures. Literature Cited 1. Rao, V. S. R.; Qasba, P. K.; Balaji, P. V. Conformation of Carbohydrates; Harwood Academic Publishers: Amsterdam, 1998. 2. Rief, M.; Oesterhelt, F.; Heymann, B.; Gaub, H. E. Science 1997, 275, 1295–1297. 3. Marszalek, P. E.; Oberhauser, A. F.; Pang, Y. -P.; Fernandez, J. M. Nature 1998, 396, 661–664. 4. Janshoff, A.; Neitzert, M.; Oberdörfer, Y.; Fuchs, H. Angew. Chem., Int. Ed. 2000, 39, 3212–3237. 5. Neumann, R. M. Biophys. J. 2002, 82, 3418–3420. 6. Haverkamp, R. G.; Williams, M. A. K.; Scott, J. E. Biomacromolecules 2005, 6, 1816–1818. 7. Haverkamp, R. G.; Marshall, A. T.; Williams, M. A. K. Phys. Rev. E 2007, 75, 021907. 8. Jackson, M. B. Molecular and Cellular Biophysics; Cambridge University Press: Cambridge, 2004. 9. Marszalek, P. E.; Pang, Y.-P.; Li, H.; Yazal, J. E.; Oberhauser, A. F.; Fernandez, J. M. Proc. Natl. Acad. Sci. USA 1999, 96, 7894–7898. 10. Oesterhelt, F.; Rief, M.; Gaub, H. E. New J. Phys. 1999, 1, 6.1–6.11. 11. Smith, E. B. Basic Chemical Thermodynamics, 5th ed; Imperial College Press: London, 2004. 12. Flory, P. J. Statistical Mechanics of Chain Molecules; Interscience Publishers: New York, 1969. 13. Wang, M. D.; Yin, H.; Landick, R.; Gelles, J.; Block, S. M. Biophys. J. 1997, 72, 1335–1346. 14. Haverkamp, R. G.; Marshall, A. T.; Williams, M. A. K. J. Phys. Chem. B 2007, 111, 13653–13657. 15. Hanke, F.; Kreuzer, H. J. Eur. Phys. J. E 2007, 22, 163–169. 16. Khatri, B. S.; Kawakami, M.; Byrne, K.; Smith, D. A.; McLeish, T. C. B. Biophys. J. 2007, 92, 1825–1835. 17. Wang, G. M.; Sevick, E. M.; Mittag, E.; Searles, D. J.; Evans, D. J. Phys. Rev. Lett. 2002, 89, 050601.
Supporting JCE Online Material
http://www.jce.divched.org/Journal/Issues/2009/Jan/abs50.html Abstract and keywords Full text (PDF) Supplement Experimental data for the stretching of dextran and hyaluronan molecules and example calculations
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