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Stochastic Chemical Kinetics Stochastic Chemical Kinetics A Special Course on Probability and Stochastic Processes for Physicists, Chemists, and Biologists Summer Term 2011 Version of July 7, 2011 Peter Schuster Institut f¨ur Theoretische Chemie der Universit¨at Wien W¨ahringerstraße 17, A-1090 Wien, Austria Phone: +43 1 4277 527 36 , Fax: +43 1 4277 527 93 E-Mail: pks @ tbi.univie.ac.at Internet: http://www.tbi.univie.ac.at/~pks 2 Peter Schuster Preface The current text contains notes that were prepared first for a course on ‘Stochastic Chemical Kinetics’ held at Vienna University in the winter term 1999/2000 and repeated in the winter term 2006/2007. The current version refers to the summer term 2011 but no claim is made that it is free of errors. The course is addressed to students of chemistry, biochemistry, molecular biology, mathematical and theoretical biology, bioinformatics, and systems biology with particular interests in phenomena observed at small particle numbers. Stochastic kinetics is an interdisciplinary subject and hence the course will contain elements from various disciplines, mainly from probability theory, mathematical statistics, stochastic processes, chemical kinetics, evo- lutionary biology, and computer science. Considerable usage of mathemat- ical language and analytical tools is indispensable, but we have consciously avoided to dwell upon deeper and more formal mathematical topics. This series of lectures will concentrate on principles rather than technical details. At the same time it will be necessary to elaborate tools that allow to treat real problems. Analytical results on stochastic processes are rare and thus it will be unavoidable to deal also with approximation methods and numerical techniques that are able to produce (approximate) results through computer calculations (see, for example, the articles [1–4]). The applica- bility of simulations to real problems depends critically on population sizes that can by handled. Present day computers can deal with 106 to 107 parti- cles, which is commonly not enough for chemical reactions but sufficient for most biological problems and accordingly the sections dealing with practical examples will contain more biological than chemical problems. The major goal of this text is to avoid distraction of the audience by taking notes and to facilitate understanding of subjects that are quite sophisticated at least in parts. At the same time the text allows for a repetition of the major issues of the course. Accordingly, an attempt was made in preparing a 3 4 Peter Schuster useful and comprehensive list of references. To study the literature in detail is recommended to every serious scholar who wants to progress towards a deeper understanding of this rather demanding discipline. Apart from a respectable number of publications mentioned during the progress of the course the following books were used in the preparation [5–9], the German textbooks [10, 11], elementary texts in English [12, 13] and in German [14]. In addition, we mention several other books on probability theory and stochastic processes [15–26]. More references will be given in the chapters on chemical and biological applications of stochastic processes. Peter Schuster Wien, March 2011. 1. History and Classical Probability An experimentalist reproduces an experiment. What can he expect to find? There are certainly limits to precision and these limits confine the repro- ducibility of experiments and at the same time restrict the predictability of outcomes. The limitations of correct predictions are commonplace: We witness them every day by watching the failures of various forecasts from the weather to the stock market. Daily experience also tells us that there is an enormous variability in the sensitivity of events with respect to preci- sion of observation. It ranges from the highly sensitive and hard to predict phenomena like the ones just mentioned to the enormous accuracy of astro- nomical predictions, for example, the precise dating of the eclipse of the sun in Europe on August 11, 1999. Most cases lie between these two extremes and careful analysis of unavoidable randomness becomes important. In this series of lectures we are heading for a formalism, which allows for proper ac- counting of the limitations of the deterministic approach and which extends the conventional description by differential equations. In order to be able to study reproducibility and sensitivity of prediction by means of a scientific approach we require a solid mathematical basis. An appropriate conceptual frame is provided by the theory of probability and stochastic processes. Conventional or deterministic variables have to be re- placed by random variables which fluctuate in the sense that different values are obtained in consecutive measurements under (within the limits of control) identical conditions. The solutions of differential equations, commonly used to describe time dependent phenomena, will not be sufficient and we shall search for proper formalisms to model stochastic processes. Fluctuations play a central role in the stochastic description of processes. The search for the origin of fluctuations is an old topic of physics, which still is of high current interest. Some examples will be mentioned in the next section 1.1. Recently, 5 6 Peter Schuster the theory of fluctuations became important because of the progress in spec- troscopic techniques, particularly in fluorescence spectroscopy. Fluctuations became directly measurable in fluorescence correlation spectroscopy [27–29]. Even single molecules can be efficiently detected, identified, and analyzed by means of these new techniques [30, 31]. In this series of lectures we shall adopt a phenomenological approach to sources of randomness or fluctuations. Thus, we shall not search here for the various mechanisms setting the limitations to reproducibility (except a short account in the next section), but we will develop a mathematical technique that allows to handle stochastic problems. In the philosophy of such a phenomenological approach the size of fluctuations, for example, is given through external parameters that can be provided by means of a deeper or more basic theory or derived from experiments. This course starts with a very brief primer of current probability theory (chapter 2), which is largely based on an undergraduate text by Kai Lai Chung [5] and an introduction to stochasticity by Hans-Otto Gregorii [11] . Then, we shall be concerned with the general formalism to describe stochastic processes (chapter 3, [6]) and present several analytical as well as numerical tools to derive or compute solutions. The following two chapters 4 and 5 deal with applications to selected problems in chemistry and biology. Stochastic Kinetics 7 1.1 Precision limits and fluctuations Conventional chemical kinetics is handling ensembles of molecules with large numbers of particles, N 1020 and more. Under many conditions1 random ≈ 4 fluctuations of particle numbers are proportional to √N. Dealing with 10− moles, being tantamount to N = 1020 particles, natural fluctuations involve 10 10 typically √N = 10 particles and thus are in the range of 10− N. Un- ± der these conditions the detection of fluctuations would require a precision 10 2 in the order of 1 : 10− , which is (almost always) impossible to achieve. Accordingly, the chemist uses concentrations rather than particle numbers, c = N/(N V ) wherein N = 6.23 1023 and V are Avogadro’s number L × L × and the volume (in dm3), respectively. Conventional chemical kinetics con- siders concentrations as continuous variables and applies deterministic meth- ods, in essence differential equations, for modeling and analysis of reactions. Thereby, it is implicitly assumed that particle numbers are sufficiently large that the limit of infinite particle numbers neglecting fluctuations is fulfilled. In 1827 the British botanist Robert Brown detected and analyzed ir- regular motions of particles in aqueous suspensions that turned out to be independent of the nature of the suspended materials – pollen grains, fine particles of glass or minerals [32]. Although Brown himself had already demonstrated that Brownian motion is not caused by some (mysterious) bi- ological effect, its origin remained kind of a riddle until Albert Einstein [33], and independently by Marian von Smoluchowski [34], published a satisfac- tory explanation in 1905 which contained two main points: (i) The motion is caused by highly frequent impacts on the pollen grain of the steadily moving molecules in the liquid in which it is suspended. (ii) The motion of the molecules in the liquid is so complicated in detail that its effect on the pollen grain can only be described probabilistically 1Computation of fluctuations and their time course of will be the subject of this course. Here we mention only that the √N law is always fulfilled in the approach towards equi- librium and stable stationary states. 2Most techniques of analytical chemistry meet serious difficulties when accuracies in 4 particle numbers of 10− or higher are required. 8 Peter Schuster in terms of frequent statistically independent impacts. In particular, Einstein showed that the number of particles per unit volume, f(x, t),3 fulfils the already known differential equation of diffusion, 2 ∂f ∂2f N exp x /(4Dt) = D with the solution f(x, t) = − , ∂t ∂x2 √4πD √t where N is the total number of particles. From the solution of the diffusion equation Einstein computes the square root of the mean square displacement, λx, the particle experiences in x-direction: 2 λx = √x¯ = √2Dt . Einstein’s treatment is based on discrete time steps and thus contains an approximation – that is well justified – but it represents the first analysis based on a probabilistic concept of a process that is comparable to the current theories and we may consider Einstein’s paper as the beginning of stochastic modeling. Brownian motion was indeed the first completely random process that became accessible to a description that was satisfactory by the standards of classical physics. Thermal motion as such had been used previously as the irregular driving force causing collisions of molecules in gases by James Clerk Maxwell and Ludwig Boltzmann.
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