But then we lose the possibility to assign probabilities for future steps. It maybe that one cannot by observations distinguish certain states, and then keep these together into larger states.

Again, an ignorance of certain details can destroy the Markovian character. If the variable we consider at one point of time is in a particular state, then there are certain probabilities to go from there to other states, and these probabilities do not depend on previous events. Step processes that have been mentioned in the previous discussions are usually regarded as Markovian. This leads to a larger scheme, but, if it provides a Markov character, it can be a substantial accomplishment. One way to simplify more general, non-Markovian processes is to include suitable extra variables.

This example shows that the neglect of some relevant variable can destroy the Markov character and, indeed, lead to a more complex process. Then, we should need further observations. If we only observe positions, this is not a Markov process, simply because we have no information about motion. Note that, however this concerns the observation of both velocity and position. Further information does not provide better prediction possibilities. If one observes the position and velocity of a Brownian particle at one time, one can predict the future motion. As it leads to relatively simple, well-defined formalisms, one usually keeps to such processes.īrownian motion is a typical example of a continuous process of Markov type. Most of the processes we describe can be assumed to be of this type. Whatever is observed before that latest observation has no influence on the outcome we next want to attain. Its result and the time lapsed since then are everything we need for assigning a probability for a new observation. For that reason, one usually tries to keep to simplified processes, still quite relevant.Ī Markov process is a process where all information that is used for predictions about the outcome at some time is given by one, latest observation. However, such a general situation becomes very cumbersome, and is almost hopeless to treat by any manageable formalism. The more observations we have made, the better we can predict the outcome at a later time. In general, that probability depends on what has been obtained in the previous observations. This means that, at each observation at a certain time, there is a certain probability to get a certain outcome. This comprises essentially everything we speak about. Basic references for this are Keizer, 1987 van Kampen, 1992 Zwanzig, 2001.Ī stochastic process means that one has a system for which there are observations at certain times, and that the outcome, that is, the observed value at each time is a random variable. We go on and now turn to stochastic processes, random variables that change with time. Understanding and modelling of biological noise, that plays an important role in cell fate decisions, environmental sensing and cell-cell communication, is another issue of main interest, as reported by Eldar and Elowitz in a recent review ( Eldar and Elowitz, 2010).Ĭlas Blomberg, in Physics of Life, 2007 18A Introduction: general account Just to cite a few advanced topics that are currently considered: models of passive and active transport in cells models of self-organization of cytoskeletal structures models for the interplay between diffusion and nonlinear chemical reactions. The relevance of noise and stochastic modelling to state-of-the-art molecular and cell biology is thus unquestionable. “The message that keeps being repeated is that the kinetics of biological processes at the intra-cellular level are stochastic, and that cellular function cannot be properly understood without building that stochasticity into in silico models” ( Wilkinson, 2012, Preface).

Stochastic processes and biological/biochemical modelling are increasingly merging in recent years as technology has started giving real insight into intra-cellular processes: quantitative real-time imaging of expression at the single-cell level and improvement in computing technology are allowing modelling and stochastic simulation of such systems at levels of detail previously impossible. Maria Francesca Carfora, in Encyclopedia of Bioinformatics and Computational Biology, 2019 Future Directions/Closing Remarks

Random processes