As a field of applied mathematics, computational biology has exploded in the last decade, and shows every sign of increasing in the next. This book overviews a few of the topics in the computational modeling of cells. I only read chapters 12 and 13 on molecular motors, and so my review will be confined to these. Nanotechnology could be described as an up-and-coming field, but in the natural world one can find examples of this technology that surpass greatly what has been accomplished by human engineers. The authors begin their articles with a few examples of natural molecular machines, including the "rotary motors" DNA helicase and bacteriophage, and the "linear motor" kinesin, the latter they refer to as a "walking enzyme". Important in the modeling of all these is the theory of stochastic processes in the guise of Brownian motion, which the authors hold is the key to understanding the mechanics of proteins. In chapter 12 they give a detailed overview of the mathematical modeling of protein dynamics, followed in chapter 13 by an illustration of the mathematical formalism in the bacterial flagellar motor, a polymerization ratchet, and a motor governing ATP synthase.
To the authors a molecular motor is an entity that converts chemical energy into mechanical force. The production of mechanical force though may involve intermediate steps of energy transduction, all these involving the release of free energy during binding events. But due to their size, molecular motors are subjected to thermal fluctuations, and thus to model their motion accurately requires the theory of stochastic processes. Thus the authors begin a study of stochastic processes, restricting their attention to ones that satisfy the Markov property. Starting with a discrete model of protein motion as a simple random walk, the authors show that the variance of the motion grows linearly with time, which is a sign of diffusive motion. The partial differential equation satisfied by the probability distribution function, in the continuous limit where the space and time scales are large enough, is left to the reader to derive as an exercise.
The authors then consider polymer growth as another example of a stochastic process, a kind of hybrid one in that it involves both discrete and continuous random variables, the position of the polymer being continuous, while the number of monomers in the polymer is discrete. The authors derive an ordinary differential equation for the probability of there being exactly n polymers at a particular time. From this they show how to obtain sample paths for polymer growth and give a brief discussion on the statistics of polymer growth.
Attention is then turned to the modeling of molecular motions, with the first example being the Brownian motion of proteins in aqueous solutions. The (stochastic) Langevin equation is given for the motion of the protein, both with and without an external force acting on the protein. To find a numerical solution of this equation is straightforward, as the authors show. But they caution however that simulation of this solution on a computer is liable to introduce spurious results, and so they derive the Smoluchowski model, a somewhat different way of looking at random motion via the evolution of ensembles of paths. In this formulation the Brownian force is replaced by a diffusion term, and the external force is modeled by a drift term.
The authors then consider the modeling of chemical reactions, which supply the energy to the molecular motors. Because of the time scales involved in these reactions, a correct treatment of them would involve quantum mechanics, but the authors use the Smoluchowski model. The simple reaction model they consider involves a positive ion binding to negatively charged amino acid, and using as reaction coordinate the distance between the ion and the amino acid, study the free energy change as a function of the reaction coordinate.
The numerical simulation of the protein motion is then considered in much greater detail, using an algorithm that preserves detailed balance. This involves converting the problem to a Markov chain and a consideration of the boundary conditions, which the authors do for the case of periodic, reflecting, and absorbing. Euler's method is used to solve the resulting equations for the Markov chain, and after dealing with issues of stability and accuracy, the Crank-Nicolson method is used. The last few sections of the chapter are devoted to the physics of these solutions and the authors give some intuitive feel for the entropic factors and energy balance on a protein motor.
In the last chapter of the book, the considerations in chapter 12 are applied to concrete molecular motors. The first one examined is a model for switching in a bacterial flagellar motor, which involves the protein CheY as a signaling pathway. The binding of CheY to the motor is modeled as a two-state process, with the binding site being either empty or occupied. The resulting set of coupled differential equations for the probabilities is solved for when the concentration of CheY is constant. An expression for the change in free energy is obtained, and the authors give a discussion of the physics in the light of what was done in the last chapter. The switching rate is computed, along with the mean first passage time.
Some other examples of molecular motors are also discussed, including the flashing racket, the polymerization ratchet, and a simplified model of the ion-driven F0 motor of ATP synthase. This latter motor is fascinating, since it describes the electrochemical energy involved in mitochondria for the production of ATP. The authors do a nice job of showing how the techniques of chapter 12 are used to solve this model, and also give an analytical solution for a certain limiting case.
