Research: Science and Education
Stochastics, the Basis of Chemical Dynamics
W
Robert de Levie Department of Chemistry, Bowdoin College, Brunswick, ME 04011;
[email protected] A repeatedly asked question is why chemical laws often have such simple mathematical expressions. Below we will consider the answer to this question. We will see that atoms, ions, and molecules do not “know” how to solve differential equations, but merely follow the statistical laws of probability. And this, as it turns out, is all it takes. To make the point, consider the decay of radioactive nuclei. We typically describe such decay in terms of either a first-order rate constant λ or a half-life time t1/2; that is, we assume that the number N of nuclear disintegrations is a simple exponential function of time, N = N0e ᎑ λt = N0 × 2᎑t/t1/2
(1)
where λ = (ln 2)/t1/2 or t1/2 = (ln 2)/ λ, and N 0 is the value of N at t = 0, just as if they follow the rate law dN/dt = ᎑ λN with Nt=0 = N0 as its initial condition. Yet when we measure the number of radioactive disintegrations with, say, a Geiger–Müller counter, we find that samples generate discrete numbers of such disintegrations; that is, N is always an integer, no matter what be the value of time t. The Geiger–Müller counter indeed counts; more specifically, it counts integers: 1, 2, 3, etc. Obviously, eq 1 cannot be completely true, because it predicts N to be a smooth, continuous function of time, whereas in fact N steps from one integer value to the next. The reason is obvious: a nucleus either has or has not disintegrated, and one clearly cannot observe, say, 31⁄4 disintegrations, since matter is quantized in atom-sized bites. Consequently, eq 1 can only be valid in a statistical sense, that is, when we have a sufficiently large number of disintegrations so that we no longer notice that both N and N0 can only assume integer values. Radioactive disintegrations are countable because they are discrete events. Such discrete events require special mathematics, couched in the language of probabilities. One can derive that, instead of eq 1, we now must solve the rate law dP0/dt = ᎑ λP0, where P0 is the probability that a radioactive nucleus has not disintegrated. This can be integrated to yield P0(t) = P0(0)e᎑λt. (The derivation can be found in appendix A of the full version of this paper, in JCE Online.W) We see that it is the probability P0 of non-disintegration, rather than the number N of non-disintegrated particles, that is a deterministic, continuous, exponential function of time. The nuclear disintegration process itself can only be described in terms of probabilities, but those probabilities follow a simple, deterministic law. And this is the true characteristic of a stochastic process: the individual occurrences may seem random, but they follow well-defined probabilistic laws, as can be established statistically by observing a sufficiently large number of such occurrences. The life-span of individual human beings is, likewise, stochastic, since it is usually not predictable, but can only be expressed in terms of probabilities. Because of these probabilities, life insurance companies can do quite well once they deal with a sufficiently large number of policyholders, because
then statistics (in the form of actuarial tables) apply. We can of course tinker with the survival probabilities by, for example, joining the army, running drugs, driving fast, or any number of other risky behaviors. Even so, there is a stochastic component to who will survive in the infantry battalion storming a fortified hill, which daredevil will get killed on the road, who will contract HIV infection, etc. Stochastic methods have long been used in the theories of physics and physical chemistry, and there are many books on this topic, such as those by McQuarrie (1) and Oppenheim et al. (2). Likewise, there have been several papers in this journal on that topic (3, 4). In this review I will instead emphasize experimental observations that directly reveal nature’s stochasticity. Opening and Closing of an Ion-Conducting Channel As a second example, consider a simple experiment reported almost two decades ago (5), in which proteins form ionconducting channels in lipid bilayer membranes, the type of membranes that are ubiquitous in nature because they surround all cells and organelles. In the experiment, the membrane separates two aqueous 0.1 M KCl solutions. The latter contain electrodes in order that a voltage may be applied to the membrane and the corresponding current recorded. Ionic solutions are many orders of magnitude more conducting than lipid bilayer membranes, so that virtually the entire applied voltage appears across the membrane. In Figure 1a we see a typical response following a step in the applied potential: the current changes almost instantaneously, then drops off in what looks like (and upon precise analysis in fact turns out to be) an exponential fashion. What we see here is the result of ion-conducting channels that were initially open, even though there was no current because the voltage applied initially was zero. When the voltage was suddenly changed to 70 mV, ions poured through those open channels, resulting in a measurable current. Soon, however, many of the ion channels started to close, because equilibrium at that voltage requires that most channels be closed. The current is directly proportional to the number of open channels, and the experiment therefore monitors the rate of channel closing. Figure 1b shows what happens when the concentration of the channel-forming protein is reduced by a factor of ten. There are now ten times fewer ion channels, the current is ten times smaller, and the signal-to-noise ratio can be expected to reflect that. And indeed, the transient is now considerably noisier. However, if we push on and dilute the protein another ten times, we note that the “noise” has a peculiar character. It does not resemble the Gaussian type of noise we so often observe in instrumental measurements at the limit of their sensitivity, but instead consists of discrete steps of constant height, as can be seen more clearly in Figure 1c. Control experiments confirm that this noise does not come from the equipment used, but is of a more fundamental nature.
JChemEd.chem.wisc.edu • Vol. 77 No. 6 June 2000 • Journal of Chemical Education
771
Research: Science and Education
Figure 1. Current–time transients observed upon applying a voltage step from 0 to 0.07 V to a lipid bilayer membrane mounted between two aqueous 0.1 M KCl solutions. One of the aqueous solutions also contains the proteinaceous material EIM (“excitability-inducing material”) in concentrations of (a) 2.0 µg/mL, (b) 0.2 µg/mL, and (c) 2 ng/mL. Note the different current scales in (a) through (c). After ref 5.
What is going on here? A small order-of-magnitude calculation can illuminate the nature of the noise. The current steps have a height of about 20 pA = 2 × 10᎑11 amperes = 2 × 10᎑11 coulombs per second. First we realize that the charge on a monovalent ion is 1.6 × 10᎑19 C. These current steps therefore correspond to the movement of about 108 ions per second, just about what one would expect from model calculations for an ion channel with a diameter of less than a nanometer and with a length of only some 30 nm, the thickness of a bilayer membrane, through which ions can travel in single file. What we witness in Figure 1c, then, is the opening and closing of single ion-permeable channels: we are looking at individual molecular events. Those are the basic events that cause the observable current. If we take a hundred times as much channel-forming protein, the current is one hundred times larger, and we must reduce the measurement sensitivity to keep the signal on scale. While the individual step size remains the same, there are now many more of them, and at the reduced sensitivity we therefore obtain an apparently smooth curve, no longer showing the true nature of the events, which in this case come in “quanta” of about 20 pA. 772
Figure 2. (a) Current–time curves of a lipid bilayer membrane between two aqueous 1 M NaCl solutions. The membrane contained a small amount of purified acetylcholine receptor, and the measurements were made 1 min after a small amount of the agonist carbamoylcholine had been added to the aqueous phase. Applied voltage: 0.1 V. The seven traces are to be “read” as lines in a book, one after the other from left to right, then from top to bottom. (b) Nine minutes later, more agonist has reached the membranebound receptors, resulting in the simultaneous openings of several acetylcholine receptor channels. After ref 6.
The mathematical description again requires that we consider probabilities because, like radioactive disintegrations, the openings and closings of ion channels are discrete, binary events: the channels are either open or closed. The new aspect here is that, whereas radioactive decay is unidirectional, opening and closing is a two-way street. A formal description then considers the rate processes 0 1, where “0” denotes a closed channel, and “1” an open one, with appropriate likelihoods λ and µ of the opening and closing of such a channel per unit time. Again we have a stochastic process, that is, one that depends on probabilities. The probabilities again follow the deterministic laws of chemical kinetics, and the likelihoods λ and µ become the usual rate constants for opening and closing channels when we observe enough of such stochastic events. The derivational details can be found in appendix D of the full paper, in JCE Online.W Figure 2 illustrates the statistical nature of channel opening and closing with another example (6 ), the opening and closing of an acetylcholine receptor channel. Acetylcholine is one of the most important neurotransmitters in the nervous system, involved both in the brain and in muscle action. Here you “see” its action close up. When an acetylcholine molecule or a similar agonist binds to its membrane-bound receptor molecule, an ion-conducting pore opens; upon subsequent dissociation of the agonist–receptor complex, the pore closes,
Journal of Chemical Education • Vol. 77 No. 6 June 2000 • JChemEd.chem.wisc.edu
Research: Science and Education
and the corresponding electrical current stops. The experiment was performed in such a way that, initially, so few agonist molecules were bound to receptors that there was only one ion-conducting channel open at one time. After more agonist had reached the membrane-bound receptors, simultaneous openings of several ion-conducting pores were observed. The stochastic nature of the process is apparent in the variable times between channel openings, as well as in the irregular time intervals during which the ion-conducting channels are open. Strictly speaking, the action observed in Figure 2 is simply the absence or presence of current flow through a pore. But, in fact, you are witnessing the binding of the agonist A to its receptor R and the subsequent dissociation of their complex AR, that is, the simple chemical reactions A + R AR, where the presence of the complex is revealed by the presence of an observable ionic current. Enzyme Kinetics Perhaps the earliest chemical example of a stochastic observation can be found in the work of Rotman (7). He studied an enzyme, β-D-galactosidase, that specifically cleaves the sugar galactose from larger molecules containing that moiety. Rotman used a synthetic substrate with little fluorescence, which the enzyme broke down into galactose and a strong fluoromer. In this way he could follow enzyme action by the appearance of fluorescence. In order to isolate single enzyme molecules, Rotman rapidly mixed 10 µL of a solution containing of about 0.9 ng of enzyme with 40 µL of a 0.12 mM substrate solution, then sprayed this mixture into a very fine mist, which was collected under silicone oil between two microscope slides. The droplets sank to the bottom of the oil and settled onto the bottom slide, retaining their spherical shape and remaining well separated, apparently as the result of electrostatic repulsion between them. Rotman then studied a large number of such droplets, all with diameters of about 14 to 15 µ m, in a thermostatted fluorescence microscope. After a number of hours, some of the droplets developed fluorescence, indicating enzyme activity. Most of those fluorescent droplets had about the same fluorescence intensity, but a small subset of them exhibited about twice as much fluorescence. This suggested that the droplets contained either zero, one, or two enzyme molecules. And that is precisely what one would expect for an enzyme having a molecular weight of about 700 kDa, at a concentration of about 0.18 ng/mL, for droplets with volumes of about 1.4 × 10᎑9 mL, which therefore contained 0.18 × 10᎑9 g/mL × 1 mol/7 × 105 g × 1.4 × 10᎑9 mL = 3.6 × 10᎑25 mol or, in view of Avogadro’s number (6 × 1023 molecules/mol), 3.6 × 10᎑25 mol × 6 × 10᎑23 molecules/mol = 0.2 molecules on average. Table 1 shows one of Rotman’s experimental results, which follow a Poisson distribution. Such an experiment can provide otherwise inaccessible information. For example, when an enzyme preparation is heated, it may lose part of its enzymatic activity. That might be due either to a decrease in the number of active enzyme molecules or to a generally diminished activity of each enzyme molecule. When Rotman did this experiment, he found that the heat treatment reduced the number of active enzyme molecules, but that the remaining still active ones had undiminished activity.
Table 1. Percentage of Droplets Appearing to Contain Various Numbers of Enzyme Molecules (7 ) Droplets
Percentage Observed
Predicted
a
Obsd Av Fluorescent Intensity
with 0 molecules
82.9
82.8
5.1
with 1 molecule
15.9
15.7
21.5
with 2 molecules
1.2
1.5
36.0
with 3 or more molecules
0
0.1
—
a
The predicted percentages were calculated from the Poisson distribution, Pn(λ t ) = (λ t )ne᎑λ t /n!, with n = 0, 1, 2, 3 and λ t = 0.189.
Nowadays, experiments like those of Rotman can be performed much more readily, with laser excitation to increase the fluorescent intensity, and regular arrays of photolithographically etched “nanoscopic vials” as sample containers (8). Why Are There So Few Stochastic Observations in Chemistry? Why is it that, in chemistry, stochastic observations are relatively rare? One reason is that the activation energies and thermodynamic energies in chemical transformations are not that much larger than the thermally available kinetic energy. In measuring radioactive decay we rely on the fact that the activation energy is very high, so that nuclear disintegration is an inherently rare event. This is why radionuclides such as 40K, with a half-life of 1.3 × 109 years, have been at it since the formation of the earth, some 5 billion years ago! Even though 40K atoms are unstable and have been falling apart since day one of the existence of our planet, that process has such a high activation energy barrier that about 7% of the original supply is still with us and has yet to disintegrate, because (1⁄2)5/1.3 ≈ 0.07. The second factor is the thermodynamic energy released as radiation upon the disintegration of an atomic nucleus. It is powerful enough to ionize a large number of atoms or molecules in, say, a Geiger–Müller counter, thereby generating a substantial signal that we can further amplify and observe. Chemical reactions don’t generate comparable energies and are therefore usually too weak to generate a signal that is clearly distinguishable from ambient experimental noise. Any successful observation of stochastic behavior in chemistry therefore requires a very specific amplification mechanism to lift the signal above the noise. In the experiments with lipid bilayers, the signal amplification derives from the ionic current through an ionpermeable channel, which converts the presence of a single channel into the movement of some hundred million ions per second, and thus into a current of the order of 10᎑11 A in the measuring circuit. In the case of phase formation through nucleation and growth, the amplification comes from the fact that the resulting cluster eventually grows to macroscopic dimensions. In the example of enzyme kinetics, the accumulating product of the enzyme-catalyzed hydrolysis, which cleaves the substrate at a rate of some 100 molecules per second, eventually builds up a macroscopically measurable fluoromer concentration. There are relatively few such builtin amplification processes in chemical kinetics.
JChemEd.chem.wisc.edu • Vol. 77 No. 6 June 2000 • Journal of Chemical Education
773
Research: Science and Education
A second reason is that most chemical experiments involve so many particles that we usually observe their statistical rather than their individual behavior. Just consider a rather small volume, say 1 µL, of a quite dilute solution, say 1 nM. Even such a minuscule amount of dissolved material, 10᎑6 × 10᎑9 = 10᎑15 moles, corresponds to an enormous number of molecules. This is a consequence of the size of Avogadro’s number, NA = 6 × 1023 molecules per mole, because the mere 10᎑15 moles of dissolved material contain 10᎑15 × 6 × 1023 = 6 × 108 molecules—that is, more than half a billion of them! Therefore, in order to observe stochastic behavior, the experimental conditions must be chosen very carefully; otherwise we will instead find the more common, deterministic response. Consequently, only processes that are quite rare, either because there are only a few participating particles (as in the case of ion channels in membranes, or the enzymes in tiny picoliter droplets) or because they are very strongly inhibited (as in radioactive decay or nucleation), will exhibit stochastic behavior. Moreover, all our examples have involved either interfacial processes per se or observations otherwise confined to two dimensions, such as to the focal plane of a microscope. This tends to make the stochastic behavior more readily observable than if it were to occur randomly in space. However, radioactivity reminds us that the interfacial aspect is merely one of observational convenience rather than of principle. Summary Fortunately, we do not need a large number of such stochastic processes to read their message; a few will do. The message is that the mathematical laws of chemical kinetics do not so much follow from differential equations as from the statistics of simple probabilities. This is one of the main reasons for what Wigner has called “the unreasonable effectiveness of mathematics in the natural sciences” (9). Molecules don’t need to compute solutions to differential equations; merely by being what they are, their encounters have inherent probabilities of making or breaking bonds, and the rest just follows. This does not mean that we can from now on do without calculus. On the contrary, it is usually much easier to compute the deterministic response from differential equations than it is to derive such a response through statistics. But we must keep in mind that in nature, we find discrete particles (atoms, molecules, ions, photons, etc.) and discrete events (such as the making or breaking of bonds). The real “laws” involved in such processes are therefore, at their core, probabilistic. Simple deterministic laws will emerge from these when we deal with sufficiently large numbers of participating particles. Are all laws of nature inherently stochastic? No, only those that deal with discrete outcomes, such as the making and breaking of chemical bonds. But that covers much of chemistry. For these, the observation of a small number of events will usually show their stochastic nature. The recent successes in isolating and observing single vapor-phase atoms,
774
as celebrated in the 1997 Nobel Prize in physics, may well lead to additional examples of stochastic behavior in chemistry. Two recent reviews provide details of such single-molecule spectroscopic observations (10, 11). In our overview we have dealt only with dynamics, that is, with change, as it can be observed in chemical kinetics. Equilibrium behavior is, of course, nothing more than a balancing act between several opposing dynamic processes, and what applies to dynamics must therefore, ipso facto, also apply to equilibria. There is, of course, a flip side to all this. Usually, chemistry deals with observations involving billions of molecules and ions, giving chemistry a statistical base that is the envy of the social sciences. The quantitative reproducibility of results in the physical sciences derives in large part from the enormous number of particles sampled. Add to this that these particles neither pretend nor deceive (as human subjects are prone to do) and you get an idea of why the deterministic “laws” of chemistry are usually obeyed so well and therefore are assigned predictive power. In that respect we are lucky that the stochastics that form the basis of chemical dynamics normally remain well hidden. Acknowledgments I am grateful to J. E. Earley and O. E. Landman of Georgetown University for helpful comments. W
Supplemental Material
The full version of this article is available in this issue of JCE Online. Literature Cited 1. McQuarrie, D. A. Stochastic Approach to Chemical Kinetics; Methuen: London, 1967. 2. Oppenheim, I.; Shuler, K. E.; Weiss, G. H. Stochastic Processes in Chemical Physics: The Master Equation; MIT Press: Cambridge, MA, 1977. 3. Boucher, E. A. J. Chem. Educ. 1974, 51, 580–584. 4. Freeman, G. R. J. Chem. Educ. 1984, 61, 944. 5. Latorre, R.; Alvarez, O. Physiol. Rev. 1981, 61, 77–150. 6. Schindler, H.; Quast, U. Proc. Natl. Acad. Sci. USA 1980, 77, 3052–3056. 7. Rotman, B. Proc. Natl. Acad. Sci. USA 1961, 47, 1981–1991. 8. Tan, W.; Yeung, E. S. Anal. Chem. 1997, 69, 4242–4248. 9. Wigner, E. P. Commun. Pure Appl. Math. 1960, 13, 1–14. 10. Trautmann, J.; Ambrose, W. P. In Single-Molecule Optical Detection, Imaging and Spectroscopy; Basché, T.; Moerner, W. E.; Orrit, M.; Wild, U. P., Eds.; VCH: Weinheim, 1997; pp 191–222. 11. Dovici, N. J.; Chen, D. D. In Single-Molecule Optical Detection, Imaging and Spectroscopy; Basché, T.; Moerner, W. E.; Orrit, M.; Wild, U. P., Eds.; VCH: Weinheim, 1997; pp 223–243.
Journal of Chemical Education • Vol. 77 No. 6 June 2000 • JChemEd.chem.wisc.edu
