Mathematical Models in the Sciences
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Molecular Life Sciences DOI 10.1007/978-1-4614-6436-5_561-1 # Springer Science+Business Media New York 2014 Mathematical Models in the Sciences John W. Cain* Department of Mathematics and Computer Science, University of Richmond, Richmond, VA, USA Synopsis A mathematical model is an attempt to describe a natural phenomenon quantitatively. Mathematical models in the molecular biosciences appear in a variety of ways: some models are deterministic while others are stochastic, some models regard time as a discrete quantity while others treat it as a continuous variable, and some models offer algebraic relationships between variables while others describe how those variables evolve over time. This entry begins with a coarse dichotomy of some of the most common types of mathematical models. As an illustration, two very different models of the same simple decay process are derived and contrasted. The entry concludes with a discussion of (i) limitations of mathematical models, (ii) validation of models against scientific data, and (iii) the iterative process of refining and improving a model. Introduction A mathematical model is an attempt to describe a natural phenomenon quantitatively. An example of a mathematical model that most students encounter during their science training is Newton’s Second Law (usually written F ¼ ma), which states that the acceleration a of an object is proportional to the net force F acting on the object, with the mass m of the object acting as the proportionality constant. As a mathematical model, Newton’s Second Law tends to be an excellent approximation for interactions of macroscopic objects that one sees in daily life. However, at a nanoscale level or at relativistic speeds, the model may perform poorly. Herein lies an important theme: models of natural systems always incorporate assumptions regarding the system being modeled. When the assump- tions are not met, the model may be a poor representation of reality, in which case successive refinements may help. Indeed, mathematical modeling (the process of constructing a model) is often an iterative process as highlighted in section “Validation, Improvement, and Limitations of Models” below. Surveying the entire spectrum of mathematical models would be far too broad of an undertaking, but the following basic distinctions are helpful in dichotomizing model types. Algebraic versus evolution equations: Model equations such as the ideal gas equation PV ¼ nRT or Newton’s Second Law express exact algebraic relationships between important variables. For example, doubling the temperature of an ideal gas in a vessel of constant volume will also double the pressure. Algebraic model equations govern what would happen if a quantity is changed, but say nothing of how the quantities change. A very different class of mathematical models fall under the heading of evolution equations: equations which describe how quantities of interest change over time. Evolution equations tend to contain derivatives (rates of change) of dependent variables with respect to time; see, for example, section “Differential Equation Model of a Decay Process” below. *Email: [email protected] Page 1 of 6 Molecular Life Sciences DOI 10.1007/978-1-4614-6436-5_561-1 # Springer Science+Business Media New York 2014 Mathematical versus statistical models: It is worth distinguishing between mathematical models and statistical models. Mathematical models are usually constructed in a more “principle- driven” manner, e.g., by appealing to Fick’s Law to describe the rate of motion of a chemical diffusing in a stationary liquid. Statistical models aim to quantify relationships between random variables – hopefully the reader will find that term sufficiently suggestive to proceed without requiring a technical definition. On some level, the distinction between mathematical and statistical models is blurred, because many of the famous equations in chemistry are derived from rather sophisticated statistical mechanics (e.g., the Arrhenius equation for dependence of reaction rates on temperature). There are two related terms worth mentioning here: deterministic versus stochastic models. Roughly speaking, a deterministic evolution model is one for which the initial state of the system completely determines all future states – randomness is not taken into account. Stochastic models do incorporate randomness, which can be important in biochemistry contexts when random interactions between molecules are important. Continuous versus discrete: Evolution equations can be subdivided into those for which time is regarded as a continuous variable and those for which it is regarded as discrete. As an illustration, consider two different phenomena relating to cardiac dynamics: (a) fluctuations in voltage across a cell membrane due to Na+,K+, and Ca2+ ion transport and (b) fluctuations in the peak voltage attained during each heartbeat. A model of the former might treat time as continuous if the goal is to use mathematics to predict voltage as a function of time (graphically rendered as a continuous trace), while a model of the latter might regard time as discrete due to the inherently discrete nature of the heartbeat. There are many instances in which it is convenient to consider all variables as varying continuously even if they are technically discrete. Concentration of a chemical in a vessel of fixed volume is a good example: technically, there are only discretely many concentrations that can be achieved since the number of molecules is an integer. Example: Two Models of a Simple Decay Process The various distinctions mentioned above can be understood through derivation of two different models of a simple decay process X ! Y. Differential Equation Model of a Decay Process An equation that contains a derivative of some dependent variable of interest is called a differential equation (DE). There are several reasons that DE models are prevalent in the sciences. First, many biological, chemical, and physical principles give rise to evolution equations which describe how important quantities change over time, as opposed to providing exact formulas for the quantities. Another advantage of DE models is that they tend to be deterministic: if a DE is supplemented with appropriate auxiliary information (such as initial conditions, which describe the state of the system at some reference time), then the future state of the system is uniquely determined by this information. Finally, there is a vast literature devoted toward the analysis of DE models, and there are standard techniques for solving (at least approximately) such equations. As the simplest illustration of where DE models arise in chemistry, consider a first-order, nonreversible decay process X ! Y in which one chemical species is converted into another. Assume that the system is closed, so that neither species is artificially added to or subtracted from the system. If x(t) denotes the number of molecules of X at time t, how might x(t) change during a short time interval of duration Dt? Provided that Dt is small, it is natural to postulate that the number of Page 2 of 6 Molecular Life Sciences DOI 10.1007/978-1-4614-6436-5_561-1 # Springer Science+Business Media New York 2014 molecules of X which spontaneously convert to Y will be proportional to both (i) the number of molecules of X and (ii) the time interval Dt. Mathematically, xtðÞþ Dt xtðÞÀkxðÞ t Dt, where k is a positive kinetic constant which would need to be measured experimentally. The expression kDt approximates the percentage of the molecules of X which are converted to Y during the time interval from t to t + Dt. As it stands, the above equation is an example of a discrete-time model – given an initial condition (IC) x(0) which measures the mass of X at an “initial” reference time t ¼ 0, the formula can be applied recursively to estimate x at times Dt,2Dt, 3Dt, and so on. Transitioning from this discrete-time model to a continuous-time DE model involves a routine procedure from introductory calculus. By preliminary algebra, xtðÞÀþ Dt xtðÞ ÀkxðÞ t , Dt after which taking the limit as Dt shrinks to 0 yields the DE dxðÞ t ¼kxðÞ t dt In words, this DE states that the rate of change of x is proportional to the amount of x present. The same DE could be used to describe a radioactive decay process, in which the rate of change of the number of radioactive atoms in a sample is proportional to the number of atoms in the sample. As DEs go, this one is very basic, a consequence of the simplicity of the underlying chemical process. The equation does not provide an exact representation of x(t), only an expression explaining how x(t)influences its own rate of change. Ideally, one wishes to produce an explicit formula for x(t) as a function of t À a formula that could be validated against experimental data and (hopefully) used for interpolation and extrapolation. Such formulas are referred to as solutions of the DE, and this DE has infinitely many solutions. To single out a specific solution of particular interest, one imposes auxiliary conditions that solutions must satisfy, the most common type being an IC. In the above DE, suppose that the initial number of molecules of X is x(0) ¼ N, a positive constant. It is possible to use standard mathematical techniques to show that the unique solution of the DE dx/dt ¼kx subject to the IC x(0) ¼ N is x(t) ¼ NeÀkt. This formula predicts simple, exponential decay of x(t) over time, agreeing with earlier intuition. A DE together with its ICs is called an initial value problem (IVP), a ubiquitous class of mathematical models in the sciences. (Stochastic) Markov Chain Model of a Decay Process The same nonreversible, first-order process X ! Y can be modeled in a probabilistic manner as follows. Let z(t) be a time-dependent random variable corresponding to the number of molecules of X that remain at time t, and let Px(t) denote the probability that z(t) ¼ x.